Documentation

Mathlib.Algebra.Group.Subgroup.Basic

Subgroups #

We prove subgroups of a group form a complete lattice, and results about images and preimages of subgroups under group homomorphisms. The bundled subgroups use bundled monoid homomorphisms.

There are also theorems about the subgroups generated by an element or a subset of a group, defined both inductively and as the infimum of the set of subgroups containing a given element/subset.

Special thanks goes to Amelia Livingston and Yury Kudryashov for their help and inspiration.

Main definitions #

Notation used here:

Definitions in the file:

Implementation notes #

Subgroup inclusion is denoted rather than , although is defined as membership of a subgroup's underlying set.

Tags #

subgroup, subgroups

theorem div_mem_comm_iff {G : Type u_1} [Group G] {S : Type u_6} {H : S} [SetLike S G] [SubgroupClass S G] {a b : G} :
a / b H b / a H
theorem sub_mem_comm_iff {G : Type u_1} [AddGroup G] {S : Type u_6} {H : S} [SetLike S G] [AddSubgroupClass S G] {a b : G} :
a - b H b - a H

Conversion to/from Additive/Multiplicative #

Subgroups of a group G are isomorphic to additive subgroups of Additive G.

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    @[simp]
    theorem Subgroup.toAddSubgroup_symm_apply_coe {G : Type u_1} [Group G] (S : AddSubgroup (Additive G)) :
    ((RelIso.symm Subgroup.toAddSubgroup) S) = Multiplicative.toAdd ⁻¹' S
    @[simp]
    theorem Subgroup.toAddSubgroup_apply_coe {G : Type u_1} [Group G] (S : Subgroup G) :
    (Subgroup.toAddSubgroup S) = Additive.toMul ⁻¹' S
    @[reducible, inline]

    Additive subgroup of an additive group Additive G are isomorphic to subgroup of G.

    Equations
    • AddSubgroup.toSubgroup' = Subgroup.toAddSubgroup.symm
    Instances For

      Additive subgroups of an additive group A are isomorphic to subgroups of Multiplicative A.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        @[simp]
        theorem AddSubgroup.toSubgroup_apply_coe {A : Type u_4} [AddGroup A] (S : AddSubgroup A) :
        (AddSubgroup.toSubgroup S) = Multiplicative.toAdd ⁻¹' S
        @[simp]
        theorem AddSubgroup.toSubgroup_symm_apply_coe {A : Type u_4} [AddGroup A] (S : Subgroup (Multiplicative A)) :
        ((RelIso.symm AddSubgroup.toSubgroup) S) = Additive.toMul ⁻¹' S
        @[reducible, inline]

        Subgroups of an additive group Multiplicative A are isomorphic to additive subgroups of A.

        Equations
        • Subgroup.toAddSubgroup' = AddSubgroup.toSubgroup.symm
        Instances For
          theorem Subgroup.div_mem_comm_iff {G : Type u_1} [Group G] (H : Subgroup G) {a b : G} :
          a / b H b / a H
          theorem AddSubgroup.sub_mem_comm_iff {G : Type u_1} [AddGroup G] (H : AddSubgroup G) {a b : G} :
          a - b H b - a H
          instance Subgroup.instTop {G : Type u_1} [Group G] :

          The subgroup G of the group G.

          Equations
          • Subgroup.instTop = { top := let __src := ; { toSubmonoid := __src, inv_mem' := } }
          instance AddSubgroup.instTop {G : Type u_1} [AddGroup G] :

          The AddSubgroup G of the AddGroup G.

          Equations
          • AddSubgroup.instTop = { top := let __src := ; { toAddSubmonoid := __src, neg_mem' := } }
          def Subgroup.topEquiv {G : Type u_1} [Group G] :
          ≃* G

          The top subgroup is isomorphic to the group.

          This is the group version of Submonoid.topEquiv.

          Equations
          • Subgroup.topEquiv = Submonoid.topEquiv
          Instances For
            def AddSubgroup.topEquiv {G : Type u_1} [AddGroup G] :
            ≃+ G

            The top additive subgroup is isomorphic to the additive group.

            This is the additive group version of AddSubmonoid.topEquiv.

            Equations
            • AddSubgroup.topEquiv = AddSubmonoid.topEquiv
            Instances For
              @[simp]
              theorem AddSubgroup.topEquiv_apply {G : Type u_1} [AddGroup G] (x : ) :
              AddSubgroup.topEquiv x = x
              @[simp]
              theorem Subgroup.topEquiv_apply {G : Type u_1} [Group G] (x : ) :
              Subgroup.topEquiv x = x
              @[simp]
              theorem Subgroup.topEquiv_symm_apply_coe {G : Type u_1} [Group G] (x : G) :
              (Subgroup.topEquiv.symm x) = x
              @[simp]
              theorem AddSubgroup.topEquiv_symm_apply_coe {G : Type u_1} [AddGroup G] (x : G) :
              (AddSubgroup.topEquiv.symm x) = x
              instance Subgroup.instBot {G : Type u_1} [Group G] :

              The trivial subgroup {1} of a group G.

              Equations
              • Subgroup.instBot = { bot := let __src := ; { toSubmonoid := __src, inv_mem' := } }
              instance AddSubgroup.instBot {G : Type u_1} [AddGroup G] :

              The trivial AddSubgroup {0} of an AddGroup G.

              Equations
              • AddSubgroup.instBot = { bot := let __src := ; { toAddSubmonoid := __src, neg_mem' := } }
              instance Subgroup.instInhabited {G : Type u_1} [Group G] :
              Equations
              • Subgroup.instInhabited = { default := }
              Equations
              • AddSubgroup.instInhabited = { default := }
              @[simp]
              theorem Subgroup.mem_bot {G : Type u_1} [Group G] {x : G} :
              x x = 1
              @[simp]
              theorem AddSubgroup.mem_bot {G : Type u_1} [AddGroup G] {x : G} :
              x x = 0
              @[simp]
              theorem Subgroup.mem_top {G : Type u_1} [Group G] (x : G) :
              @[simp]
              theorem AddSubgroup.mem_top {G : Type u_1} [AddGroup G] (x : G) :
              @[simp]
              theorem Subgroup.coe_top {G : Type u_1} [Group G] :
              = Set.univ
              @[simp]
              theorem AddSubgroup.coe_top {G : Type u_1} [AddGroup G] :
              = Set.univ
              @[simp]
              theorem Subgroup.coe_bot {G : Type u_1} [Group G] :
              = {1}
              @[simp]
              theorem AddSubgroup.coe_bot {G : Type u_1} [AddGroup G] :
              = {0}
              Equations
              • Subgroup.instUniqueSubtypeMemBot = { default := 1, uniq := }
              Equations
              • AddSubgroup.instUniqueSubtypeMemBot = { default := 0, uniq := }
              @[simp]
              theorem Subgroup.top_toSubmonoid {G : Type u_1} [Group G] :
              .toSubmonoid =
              @[simp]
              theorem AddSubgroup.top_toAddSubmonoid {G : Type u_1} [AddGroup G] :
              .toAddSubmonoid =
              @[simp]
              theorem Subgroup.bot_toSubmonoid {G : Type u_1} [Group G] :
              .toSubmonoid =
              @[simp]
              theorem AddSubgroup.bot_toAddSubmonoid {G : Type u_1} [AddGroup G] :
              .toAddSubmonoid =
              theorem Subgroup.eq_bot_iff_forall {G : Type u_1} [Group G] (H : Subgroup G) :
              H = xH, x = 1
              theorem AddSubgroup.eq_bot_iff_forall {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
              H = xH, x = 0
              @[simp]
              theorem Subgroup.coe_eq_univ {G : Type u_1} [Group G] {H : Subgroup G} :
              H = Set.univ H =
              @[simp]
              theorem AddSubgroup.coe_eq_univ {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
              H = Set.univ H =
              theorem Subgroup.coe_eq_singleton {G : Type u_1} [Group G] {H : Subgroup G} :
              (∃ (g : G), H = {g}) H =
              theorem AddSubgroup.coe_eq_singleton {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
              (∃ (g : G), H = {g}) H =
              theorem Subgroup.nontrivial_iff_exists_ne_one {G : Type u_1} [Group G] (H : Subgroup G) :
              Nontrivial H xH, x 1
              theorem AddSubgroup.nontrivial_iff_exists_ne_zero {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
              Nontrivial H xH, x 0
              theorem Subgroup.exists_ne_one_of_nontrivial {G : Type u_1} [Group G] (H : Subgroup G) [Nontrivial H] :
              xH, x 1
              theorem AddSubgroup.exists_ne_zero_of_nontrivial {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [Nontrivial H] :
              xH, x 0
              theorem Subgroup.bot_or_nontrivial {G : Type u_1} [Group G] (H : Subgroup G) :

              A subgroup is either the trivial subgroup or nontrivial.

              A subgroup is either the trivial subgroup or nontrivial.

              theorem Subgroup.bot_or_exists_ne_one {G : Type u_1} [Group G] (H : Subgroup G) :
              H = xH, x 1

              A subgroup is either the trivial subgroup or contains a non-identity element.

              theorem AddSubgroup.bot_or_exists_ne_zero {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
              H = xH, x 0

              A subgroup is either the trivial subgroup or contains a nonzero element.

              theorem Subgroup.ne_bot_iff_exists_ne_one {G : Type u_1} [Group G] {H : Subgroup G} :
              H ∃ (a : H), a 1
              theorem AddSubgroup.ne_bot_iff_exists_ne_zero {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
              H ∃ (a : H), a 0
              instance Subgroup.instMin {G : Type u_1} [Group G] :

              The inf of two subgroups is their intersection.

              Equations
              • Subgroup.instMin = { min := fun (H₁ H₂ : Subgroup G) => let __src := H₁.toSubmonoid H₂.toSubmonoid; { toSubmonoid := __src, inv_mem' := } }
              instance AddSubgroup.instMin {G : Type u_1} [AddGroup G] :

              The inf of two AddSubgroups is their intersection.

              Equations
              • AddSubgroup.instMin = { min := fun (H₁ H₂ : AddSubgroup G) => let __src := H₁.toAddSubmonoid H₂.toAddSubmonoid; { toAddSubmonoid := __src, neg_mem' := } }
              @[simp]
              theorem Subgroup.coe_inf {G : Type u_1} [Group G] (p p' : Subgroup G) :
              (p p') = p p'
              @[simp]
              theorem AddSubgroup.coe_inf {G : Type u_1} [AddGroup G] (p p' : AddSubgroup G) :
              (p p') = p p'
              @[simp]
              theorem Subgroup.mem_inf {G : Type u_1} [Group G] {p p' : Subgroup G} {x : G} :
              x p p' x p x p'
              @[simp]
              theorem AddSubgroup.mem_inf {G : Type u_1} [AddGroup G] {p p' : AddSubgroup G} {x : G} :
              x p p' x p x p'
              instance Subgroup.instInfSet {G : Type u_1} [Group G] :
              Equations
              • Subgroup.instInfSet = { sInf := fun (s : Set (Subgroup G)) => let __src := (⨅ Ss, S.toSubmonoid).copy (⋂ Ss, S) ; { toSubmonoid := __src, inv_mem' := } }
              Equations
              • AddSubgroup.instInfSet = { sInf := fun (s : Set (AddSubgroup G)) => let __src := (⨅ Ss, S.toAddSubmonoid).copy (⋂ Ss, S) ; { toAddSubmonoid := __src, neg_mem' := } }
              @[simp]
              theorem Subgroup.coe_sInf {G : Type u_1} [Group G] (H : Set (Subgroup G)) :
              (sInf H) = sH, s
              @[simp]
              theorem AddSubgroup.coe_sInf {G : Type u_1} [AddGroup G] (H : Set (AddSubgroup G)) :
              (sInf H) = sH, s
              @[simp]
              theorem Subgroup.mem_sInf {G : Type u_1} [Group G] {S : Set (Subgroup G)} {x : G} :
              x sInf S pS, x p
              @[simp]
              theorem AddSubgroup.mem_sInf {G : Type u_1} [AddGroup G] {S : Set (AddSubgroup G)} {x : G} :
              x sInf S pS, x p
              theorem Subgroup.mem_iInf {G : Type u_1} [Group G] {ι : Sort u_5} {S : ιSubgroup G} {x : G} :
              x ⨅ (i : ι), S i ∀ (i : ι), x S i
              theorem AddSubgroup.mem_iInf {G : Type u_1} [AddGroup G] {ι : Sort u_5} {S : ιAddSubgroup G} {x : G} :
              x ⨅ (i : ι), S i ∀ (i : ι), x S i
              @[simp]
              theorem Subgroup.coe_iInf {G : Type u_1} [Group G] {ι : Sort u_5} {S : ιSubgroup G} :
              (⨅ (i : ι), S i) = ⋂ (i : ι), (S i)
              @[simp]
              theorem AddSubgroup.coe_iInf {G : Type u_1} [AddGroup G] {ι : Sort u_5} {S : ιAddSubgroup G} :
              (⨅ (i : ι), S i) = ⋂ (i : ι), (S i)

              Subgroups of a group form a complete lattice.

              Equations

              The AddSubgroups of an AddGroup form a complete lattice.

              Equations
              theorem Subgroup.mem_sup_left {G : Type u_1} [Group G] {S T : Subgroup G} {x : G} :
              x Sx S T
              theorem AddSubgroup.mem_sup_left {G : Type u_1} [AddGroup G] {S T : AddSubgroup G} {x : G} :
              x Sx S T
              theorem Subgroup.mem_sup_right {G : Type u_1} [Group G] {S T : Subgroup G} {x : G} :
              x Tx S T
              theorem AddSubgroup.mem_sup_right {G : Type u_1} [AddGroup G] {S T : AddSubgroup G} {x : G} :
              x Tx S T
              theorem Subgroup.mul_mem_sup {G : Type u_1} [Group G] {S T : Subgroup G} {x y : G} (hx : x S) (hy : y T) :
              x * y S T
              theorem AddSubgroup.add_mem_sup {G : Type u_1} [AddGroup G] {S T : AddSubgroup G} {x y : G} (hx : x S) (hy : y T) :
              x + y S T
              theorem Subgroup.mem_iSup_of_mem {G : Type u_1} [Group G] {ι : Sort u_5} {S : ιSubgroup G} (i : ι) {x : G} :
              x S ix iSup S
              theorem AddSubgroup.mem_iSup_of_mem {G : Type u_1} [AddGroup G] {ι : Sort u_5} {S : ιAddSubgroup G} (i : ι) {x : G} :
              x S ix iSup S
              theorem Subgroup.mem_sSup_of_mem {G : Type u_1} [Group G] {S : Set (Subgroup G)} {s : Subgroup G} (hs : s S) {x : G} :
              x sx sSup S
              theorem AddSubgroup.mem_sSup_of_mem {G : Type u_1} [AddGroup G] {S : Set (AddSubgroup G)} {s : AddSubgroup G} (hs : s S) {x : G} :
              x sx sSup S
              Equations
              • Subgroup.instUniqueOfSubsingleton = { default := , uniq := }
              Equations
              • AddSubgroup.instUniqueOfSubsingleton = { default := , uniq := }
              Equations
              • =
              Equations
              • =
              theorem Subgroup.eq_top_iff' {G : Type u_1} [Group G] (H : Subgroup G) :
              H = ∀ (x : G), x H
              theorem AddSubgroup.eq_top_iff' {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
              H = ∀ (x : G), x H
              def Subgroup.closure {G : Type u_1} [Group G] (k : Set G) :

              The Subgroup generated by a set.

              Equations
              Instances For
                def AddSubgroup.closure {G : Type u_1} [AddGroup G] (k : Set G) :

                The AddSubgroup generated by a set

                Equations
                Instances For
                  theorem Subgroup.mem_closure {G : Type u_1} [Group G] {k : Set G} {x : G} :
                  x Subgroup.closure k ∀ (K : Subgroup G), k Kx K
                  theorem AddSubgroup.mem_closure {G : Type u_1} [AddGroup G] {k : Set G} {x : G} :
                  x AddSubgroup.closure k ∀ (K : AddSubgroup G), k Kx K
                  @[simp]
                  theorem Subgroup.subset_closure {G : Type u_1} [Group G] {k : Set G} :

                  The subgroup generated by a set includes the set.

                  @[simp]
                  theorem AddSubgroup.subset_closure {G : Type u_1} [AddGroup G] {k : Set G} :

                  The AddSubgroup generated by a set includes the set.

                  theorem Subgroup.not_mem_of_not_mem_closure {G : Type u_1} [Group G] {k : Set G} {P : G} (hP : PSubgroup.closure k) :
                  Pk
                  theorem AddSubgroup.not_mem_of_not_mem_closure {G : Type u_1} [AddGroup G] {k : Set G} {P : G} (hP : PAddSubgroup.closure k) :
                  Pk
                  @[simp]
                  theorem Subgroup.closure_le {G : Type u_1} [Group G] (K : Subgroup G) {k : Set G} :

                  A subgroup K includes closure k if and only if it includes k.

                  @[simp]
                  theorem AddSubgroup.closure_le {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {k : Set G} :

                  An additive subgroup K includes closure k if and only if it includes k

                  theorem Subgroup.closure_eq_of_le {G : Type u_1} [Group G] (K : Subgroup G) {k : Set G} (h₁ : k K) (h₂ : K Subgroup.closure k) :
                  theorem AddSubgroup.closure_eq_of_le {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {k : Set G} (h₁ : k K) (h₂ : K AddSubgroup.closure k) :
                  theorem Subgroup.closure_induction {G : Type u_1} [Group G] {k : Set G} {p : (g : G) → g Subgroup.closure kProp} (mem : ∀ (x : G) (hx : x k), p x ) (one : p 1 ) (mul : ∀ (x y : G) (hx : x Subgroup.closure k) (hy : y Subgroup.closure k), p x hxp y hyp (x * y) ) (inv : ∀ (x : G) (hx : x Subgroup.closure k), p x hxp x⁻¹ ) {x : G} (hx : x Subgroup.closure k) :
                  p x hx

                  An induction principle for closure membership. If p holds for 1 and all elements of k, and is preserved under multiplication and inverse, then p holds for all elements of the closure of k.

                  See also Subgroup.closure_induction_left and Subgroup.closure_induction_right for versions that only require showing p is preserved by multiplication by elements in k.

                  theorem AddSubgroup.closure_induction {G : Type u_1} [AddGroup G] {k : Set G} {p : (g : G) → g AddSubgroup.closure kProp} (mem : ∀ (x : G) (hx : x k), p x ) (one : p 0 ) (mul : ∀ (x y : G) (hx : x AddSubgroup.closure k) (hy : y AddSubgroup.closure k), p x hxp y hyp (x + y) ) (inv : ∀ (x : G) (hx : x AddSubgroup.closure k), p x hxp (-x) ) {x : G} (hx : x AddSubgroup.closure k) :
                  p x hx

                  An induction principle for additive closure membership. If p holds for 0 and all elements of k, and is preserved under addition and inverses, then p holds for all elements of the additive closure of k.

                  See also AddSubgroup.closure_induction_left and AddSubgroup.closure_induction_left for versions that only require showing p is preserved by addition by elements in k.

                  @[deprecated Subgroup.closure_induction]
                  theorem Subgroup.closure_induction' {G : Type u_1} [Group G] {k : Set G} {p : (g : G) → g Subgroup.closure kProp} (mem : ∀ (x : G) (hx : x k), p x ) (one : p 1 ) (mul : ∀ (x y : G) (hx : x Subgroup.closure k) (hy : y Subgroup.closure k), p x hxp y hyp (x * y) ) (inv : ∀ (x : G) (hx : x Subgroup.closure k), p x hxp x⁻¹ ) {x : G} (hx : x Subgroup.closure k) :
                  p x hx

                  Alias of Subgroup.closure_induction.


                  An induction principle for closure membership. If p holds for 1 and all elements of k, and is preserved under multiplication and inverse, then p holds for all elements of the closure of k.

                  See also Subgroup.closure_induction_left and Subgroup.closure_induction_right for versions that only require showing p is preserved by multiplication by elements in k.

                  theorem Subgroup.closure_induction₂ {G : Type u_1} [Group G] {k : Set G} {p : (x y : G) → x Subgroup.closure ky Subgroup.closure kProp} (mem : ∀ (x y : G) (hx : x k) (hy : y k), p x y ) (one_left : ∀ (x : G) (hx : x Subgroup.closure k), p 1 x hx) (one_right : ∀ (x : G) (hx : x Subgroup.closure k), p x 1 hx ) (mul_left : ∀ (x y z : G) (hx : x Subgroup.closure k) (hy : y Subgroup.closure k) (hz : z Subgroup.closure k), p x z hx hzp y z hy hzp (x * y) z hz) (mul_right : ∀ (y z x : G) (hy : y Subgroup.closure k) (hz : z Subgroup.closure k) (hx : x Subgroup.closure k), p x y hx hyp x z hx hzp x (y * z) hx ) (inv_left : ∀ (x y : G) (hx : x Subgroup.closure k) (hy : y Subgroup.closure k), p x y hx hyp x⁻¹ y hy) (inv_right : ∀ (x y : G) (hx : x Subgroup.closure k) (hy : y Subgroup.closure k), p x y hx hyp x y⁻¹ hx ) {x y : G} (hx : x Subgroup.closure k) (hy : y Subgroup.closure k) :
                  p x y hx hy

                  An induction principle for closure membership for predicates with two arguments.

                  theorem AddSubgroup.closure_induction₂ {G : Type u_1} [AddGroup G] {k : Set G} {p : (x y : G) → x AddSubgroup.closure ky AddSubgroup.closure kProp} (mem : ∀ (x y : G) (hx : x k) (hy : y k), p x y ) (one_left : ∀ (x : G) (hx : x AddSubgroup.closure k), p 0 x hx) (one_right : ∀ (x : G) (hx : x AddSubgroup.closure k), p x 0 hx ) (mul_left : ∀ (x y z : G) (hx : x AddSubgroup.closure k) (hy : y AddSubgroup.closure k) (hz : z AddSubgroup.closure k), p x z hx hzp y z hy hzp (x + y) z hz) (mul_right : ∀ (y z x : G) (hy : y AddSubgroup.closure k) (hz : z AddSubgroup.closure k) (hx : x AddSubgroup.closure k), p x y hx hyp x z hx hzp x (y + z) hx ) (inv_left : ∀ (x y : G) (hx : x AddSubgroup.closure k) (hy : y AddSubgroup.closure k), p x y hx hyp (-x) y hy) (inv_right : ∀ (x y : G) (hx : x AddSubgroup.closure k) (hy : y AddSubgroup.closure k), p x y hx hyp x (-y) hx ) {x y : G} (hx : x AddSubgroup.closure k) (hy : y AddSubgroup.closure k) :
                  p x y hx hy

                  An induction principle for additive closure membership, for predicates with two arguments.

                  @[simp]
                  theorem Subgroup.closure_closure_coe_preimage {G : Type u_1} [Group G] {k : Set G} :
                  Subgroup.closure (Subtype.val ⁻¹' k) =
                  @[simp]
                  def Subgroup.closureCommGroupOfComm {G : Type u_1} [Group G] {k : Set G} (hcomm : xk, yk, x * y = y * x) :

                  If all the elements of a set s commute, then closure s is a commutative group.

                  Equations
                  Instances For
                    def AddSubgroup.closureAddCommGroupOfComm {G : Type u_1} [AddGroup G] {k : Set G} (hcomm : xk, yk, x + y = y + x) :

                    If all the elements of a set s commute, then closure s is an additive commutative group.

                    Equations
                    Instances For
                      def Subgroup.gi (G : Type u_1) [Group G] :
                      GaloisInsertion Subgroup.closure SetLike.coe

                      closure forms a Galois insertion with the coercion to set.

                      Equations
                      Instances For
                        def AddSubgroup.gi (G : Type u_1) [AddGroup G] :
                        GaloisInsertion AddSubgroup.closure SetLike.coe

                        closure forms a Galois insertion with the coercion to set.

                        Equations
                        Instances For
                          theorem Subgroup.closure_mono {G : Type u_1} [Group G] ⦃h k : Set G (h' : h k) :

                          Subgroup closure of a set is monotone in its argument: if h ⊆ k, then closure h ≤ closure k.

                          theorem AddSubgroup.closure_mono {G : Type u_1} [AddGroup G] ⦃h k : Set G (h' : h k) :

                          Additive subgroup closure of a set is monotone in its argument: if h ⊆ k, then closure h ≤ closure k

                          @[simp]
                          theorem Subgroup.closure_eq {G : Type u_1} [Group G] (K : Subgroup G) :

                          Closure of a subgroup K equals K.

                          @[simp]

                          Additive closure of an additive subgroup K equals K

                          @[simp]
                          theorem Subgroup.closure_univ {G : Type u_1} [Group G] :
                          @[simp]
                          theorem Subgroup.sup_eq_closure {G : Type u_1} [Group G] (H H' : Subgroup G) :
                          H H' = Subgroup.closure (H H')
                          theorem AddSubgroup.sup_eq_closure {G : Type u_1} [AddGroup G] (H H' : AddSubgroup G) :
                          H H' = AddSubgroup.closure (H H')
                          theorem Subgroup.closure_iUnion {G : Type u_1} [Group G] {ι : Sort u_5} (s : ιSet G) :
                          Subgroup.closure (⋃ (i : ι), s i) = ⨆ (i : ι), Subgroup.closure (s i)
                          theorem AddSubgroup.closure_iUnion {G : Type u_1} [AddGroup G] {ι : Sort u_5} (s : ιSet G) :
                          AddSubgroup.closure (⋃ (i : ι), s i) = ⨆ (i : ι), AddSubgroup.closure (s i)
                          @[simp]
                          theorem Subgroup.closure_eq_bot_iff {G : Type u_1} [Group G] {k : Set G} :
                          @[simp]
                          theorem Subgroup.iSup_eq_closure {G : Type u_1} [Group G] {ι : Sort u_5} (p : ιSubgroup G) :
                          ⨆ (i : ι), p i = Subgroup.closure (⋃ (i : ι), (p i))
                          theorem AddSubgroup.iSup_eq_closure {G : Type u_1} [AddGroup G] {ι : Sort u_5} (p : ιAddSubgroup G) :
                          ⨆ (i : ι), p i = AddSubgroup.closure (⋃ (i : ι), (p i))
                          theorem Subgroup.mem_closure_singleton {G : Type u_1} [Group G] {x y : G} :
                          y Subgroup.closure {x} ∃ (n : ), x ^ n = y

                          The subgroup generated by an element of a group equals the set of integer number powers of the element.

                          theorem AddSubgroup.mem_closure_singleton {G : Type u_1} [AddGroup G] {x y : G} :
                          y AddSubgroup.closure {x} ∃ (n : ), n x = y

                          The AddSubgroup generated by an element of an AddGroup equals the set of natural number multiples of the element.

                          @[simp]
                          theorem Subgroup.mem_iSup_of_directed {G : Type u_1} [Group G] {ι : Sort u_5} [hι : Nonempty ι] {K : ιSubgroup G} (hK : Directed (fun (x1 x2 : Subgroup G) => x1 x2) K) {x : G} :
                          x iSup K ∃ (i : ι), x K i
                          theorem AddSubgroup.mem_iSup_of_directed {G : Type u_1} [AddGroup G] {ι : Sort u_5} [hι : Nonempty ι] {K : ιAddSubgroup G} (hK : Directed (fun (x1 x2 : AddSubgroup G) => x1 x2) K) {x : G} :
                          x iSup K ∃ (i : ι), x K i
                          theorem Subgroup.coe_iSup_of_directed {G : Type u_1} [Group G] {ι : Sort u_5} [Nonempty ι] {S : ιSubgroup G} (hS : Directed (fun (x1 x2 : Subgroup G) => x1 x2) S) :
                          (⨆ (i : ι), S i) = ⋃ (i : ι), (S i)
                          theorem AddSubgroup.coe_iSup_of_directed {G : Type u_1} [AddGroup G] {ι : Sort u_5} [Nonempty ι] {S : ιAddSubgroup G} (hS : Directed (fun (x1 x2 : AddSubgroup G) => x1 x2) S) :
                          (⨆ (i : ι), S i) = ⋃ (i : ι), (S i)
                          theorem Subgroup.mem_sSup_of_directedOn {G : Type u_1} [Group G] {K : Set (Subgroup G)} (Kne : K.Nonempty) (hK : DirectedOn (fun (x1 x2 : Subgroup G) => x1 x2) K) {x : G} :
                          x sSup K sK, x s
                          theorem AddSubgroup.mem_sSup_of_directedOn {G : Type u_1} [AddGroup G] {K : Set (AddSubgroup G)} (Kne : K.Nonempty) (hK : DirectedOn (fun (x1 x2 : AddSubgroup G) => x1 x2) K) {x : G} :
                          x sSup K sK, x s
                          def Subgroup.comap {G : Type u_1} [Group G] {N : Type u_7} [Group N] (f : G →* N) (H : Subgroup N) :

                          The preimage of a subgroup along a monoid homomorphism is a subgroup.

                          Equations
                          Instances For
                            def AddSubgroup.comap {G : Type u_1} [AddGroup G] {N : Type u_7} [AddGroup N] (f : G →+ N) (H : AddSubgroup N) :

                            The preimage of an AddSubgroup along an AddMonoid homomorphism is an AddSubgroup.

                            Equations
                            Instances For
                              @[simp]
                              theorem Subgroup.coe_comap {G : Type u_1} [Group G] {N : Type u_5} [Group N] (K : Subgroup N) (f : G →* N) :
                              (Subgroup.comap f K) = f ⁻¹' K
                              @[simp]
                              theorem AddSubgroup.coe_comap {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (K : AddSubgroup N) (f : G →+ N) :
                              (AddSubgroup.comap f K) = f ⁻¹' K
                              @[simp]
                              theorem Subgroup.toAddSubgroup_comap {G : Type u_1} [Group G] {G₂ : Type u_7} [Group G₂] (f : G →* G₂) (s : Subgroup G₂) :
                              AddSubgroup.comap (MonoidHom.toAdditive f) (Subgroup.toAddSubgroup s) = Subgroup.toAddSubgroup (Subgroup.comap f s)
                              @[simp]
                              theorem AddSubgroup.toSubgroup_comap {A : Type u_7} {A₂ : Type u_8} [AddGroup A] [AddGroup A₂] (f : A →+ A₂) (s : AddSubgroup A₂) :
                              Subgroup.comap (AddMonoidHom.toMultiplicative f) (AddSubgroup.toSubgroup s) = AddSubgroup.toSubgroup (AddSubgroup.comap f s)
                              @[simp]
                              theorem Subgroup.mem_comap {G : Type u_1} [Group G] {N : Type u_5} [Group N] {K : Subgroup N} {f : G →* N} {x : G} :
                              @[simp]
                              theorem AddSubgroup.mem_comap {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {K : AddSubgroup N} {f : G →+ N} {x : G} :
                              theorem Subgroup.comap_mono {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K K' : Subgroup N} :
                              theorem AddSubgroup.comap_mono {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K K' : AddSubgroup N} :
                              theorem Subgroup.comap_comap {G : Type u_1} [Group G] {N : Type u_5} [Group N] {P : Type u_6} [Group P] (K : Subgroup P) (g : N →* P) (f : G →* N) :
                              theorem AddSubgroup.comap_comap {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {P : Type u_6} [AddGroup P] (K : AddSubgroup P) (g : N →+ P) (f : G →+ N) :
                              @[simp]
                              theorem Subgroup.comap_id {N : Type u_5} [Group N] (K : Subgroup N) :
                              def Subgroup.map {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (H : Subgroup G) :

                              The image of a subgroup along a monoid homomorphism is a subgroup.

                              Equations
                              • Subgroup.map f H = { carrier := f '' H, mul_mem' := , one_mem' := , inv_mem' := }
                              Instances For
                                def AddSubgroup.map {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup G) :

                                The image of an AddSubgroup along an AddMonoid homomorphism is an AddSubgroup.

                                Equations
                                • AddSubgroup.map f H = { carrier := f '' H, add_mem' := , zero_mem' := , neg_mem' := }
                                Instances For
                                  @[simp]
                                  theorem Subgroup.coe_map {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (K : Subgroup G) :
                                  (Subgroup.map f K) = f '' K
                                  @[simp]
                                  theorem AddSubgroup.coe_map {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (K : AddSubgroup G) :
                                  (AddSubgroup.map f K) = f '' K
                                  @[simp]
                                  theorem Subgroup.mem_map {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K : Subgroup G} {y : N} :
                                  y Subgroup.map f K xK, f x = y
                                  @[simp]
                                  theorem AddSubgroup.mem_map {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K : AddSubgroup G} {y : N} :
                                  y AddSubgroup.map f K xK, f x = y
                                  theorem Subgroup.mem_map_of_mem {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) {K : Subgroup G} {x : G} (hx : x K) :
                                  theorem AddSubgroup.mem_map_of_mem {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) {K : AddSubgroup G} {x : G} (hx : x K) :
                                  theorem Subgroup.apply_coe_mem_map {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (K : Subgroup G) (x : K) :
                                  f x Subgroup.map f K
                                  theorem AddSubgroup.apply_coe_mem_map {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (K : AddSubgroup G) (x : K) :
                                  f x AddSubgroup.map f K
                                  theorem Subgroup.map_mono {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K K' : Subgroup G} :
                                  K K'Subgroup.map f K Subgroup.map f K'
                                  theorem AddSubgroup.map_mono {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K K' : AddSubgroup G} :
                                  @[simp]
                                  theorem Subgroup.map_id {G : Type u_1} [Group G] (K : Subgroup G) :
                                  @[simp]
                                  theorem Subgroup.map_map {G : Type u_1} [Group G] (K : Subgroup G) {N : Type u_5} [Group N] {P : Type u_6} [Group P] (g : N →* P) (f : G →* N) :
                                  theorem AddSubgroup.map_map {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {N : Type u_5} [AddGroup N] {P : Type u_6} [AddGroup P] (g : N →+ P) (f : G →+ N) :
                                  @[simp]
                                  theorem Subgroup.map_one_eq_bot {G : Type u_1} [Group G] (K : Subgroup G) {N : Type u_5} [Group N] :
                                  @[simp]
                                  theorem AddSubgroup.map_zero_eq_bot {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {N : Type u_5} [AddGroup N] :
                                  theorem Subgroup.mem_map_equiv {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G ≃* N} {K : Subgroup G} {x : N} :
                                  x Subgroup.map f.toMonoidHom K f.symm x K
                                  theorem AddSubgroup.mem_map_equiv {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G ≃+ N} {K : AddSubgroup G} {x : N} :
                                  x AddSubgroup.map f.toAddMonoidHom K f.symm x K
                                  @[simp]
                                  theorem Subgroup.mem_map_iff_mem {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (hf : Function.Injective f) {K : Subgroup G} {x : G} :
                                  f x Subgroup.map f K x K
                                  @[simp]
                                  theorem AddSubgroup.mem_map_iff_mem {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (hf : Function.Injective f) {K : AddSubgroup G} {x : G} :
                                  theorem Subgroup.map_equiv_eq_comap_symm' {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G ≃* N) (K : Subgroup G) :
                                  Subgroup.map f.toMonoidHom K = Subgroup.comap f.symm.toMonoidHom K
                                  theorem AddSubgroup.map_equiv_eq_comap_symm' {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G ≃+ N) (K : AddSubgroup G) :
                                  AddSubgroup.map f.toAddMonoidHom K = AddSubgroup.comap f.symm.toAddMonoidHom K
                                  theorem Subgroup.map_equiv_eq_comap_symm {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G ≃* N) (K : Subgroup G) :
                                  Subgroup.map (↑f) K = Subgroup.comap (↑f.symm) K
                                  theorem AddSubgroup.map_equiv_eq_comap_symm {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G ≃+ N) (K : AddSubgroup G) :
                                  AddSubgroup.map (↑f) K = AddSubgroup.comap (↑f.symm) K
                                  theorem Subgroup.comap_equiv_eq_map_symm {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : N ≃* G) (K : Subgroup G) :
                                  Subgroup.comap (↑f) K = Subgroup.map (↑f.symm) K
                                  theorem AddSubgroup.comap_equiv_eq_map_symm {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : N ≃+ G) (K : AddSubgroup G) :
                                  AddSubgroup.comap (↑f) K = AddSubgroup.map (↑f.symm) K
                                  theorem Subgroup.comap_equiv_eq_map_symm' {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : N ≃* G) (K : Subgroup G) :
                                  Subgroup.comap f.toMonoidHom K = Subgroup.map f.symm.toMonoidHom K
                                  theorem AddSubgroup.comap_equiv_eq_map_symm' {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : N ≃+ G) (K : AddSubgroup G) :
                                  AddSubgroup.comap f.toAddMonoidHom K = AddSubgroup.map f.symm.toAddMonoidHom K
                                  theorem Subgroup.map_symm_eq_iff_map_eq {G : Type u_1} [Group G] (K : Subgroup G) {N : Type u_5} [Group N] {H : Subgroup N} {e : G ≃* N} :
                                  Subgroup.map (↑e.symm) H = K Subgroup.map (↑e) K = H
                                  theorem AddSubgroup.map_symm_eq_iff_map_eq {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {N : Type u_5} [AddGroup N] {H : AddSubgroup N} {e : G ≃+ N} :
                                  AddSubgroup.map (↑e.symm) H = K AddSubgroup.map (↑e) K = H
                                  theorem Subgroup.map_le_iff_le_comap {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K : Subgroup G} {H : Subgroup N} :
                                  theorem AddSubgroup.map_le_iff_le_comap {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K : AddSubgroup G} {H : AddSubgroup N} :
                                  theorem Subgroup.gc_map_comap {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                  theorem Subgroup.map_sup {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup G) (f : G →* N) :
                                  theorem AddSubgroup.map_sup {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N) :
                                  theorem Subgroup.map_iSup {G : Type u_1} [Group G] {N : Type u_5} [Group N] {ι : Sort u_7} (f : G →* N) (s : ιSubgroup G) :
                                  Subgroup.map f (iSup s) = ⨆ (i : ι), Subgroup.map f (s i)
                                  theorem AddSubgroup.map_iSup {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {ι : Sort u_7} (f : G →+ N) (s : ιAddSubgroup G) :
                                  AddSubgroup.map f (iSup s) = ⨆ (i : ι), AddSubgroup.map f (s i)
                                  theorem Subgroup.map_inf {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup G) (f : G →* N) (hf : Function.Injective f) :
                                  theorem AddSubgroup.map_inf {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N) (hf : Function.Injective f) :
                                  theorem Subgroup.map_iInf {G : Type u_1} [Group G] {N : Type u_5} [Group N] {ι : Sort u_7} [Nonempty ι] (f : G →* N) (hf : Function.Injective f) (s : ιSubgroup G) :
                                  Subgroup.map f (iInf s) = ⨅ (i : ι), Subgroup.map f (s i)
                                  theorem AddSubgroup.map_iInf {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {ι : Sort u_7} [Nonempty ι] (f : G →+ N) (hf : Function.Injective f) (s : ιAddSubgroup G) :
                                  AddSubgroup.map f (iInf s) = ⨅ (i : ι), AddSubgroup.map f (s i)
                                  theorem Subgroup.comap_sup_comap_le {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup N) (f : G →* N) :
                                  theorem Subgroup.iSup_comap_le {G : Type u_1} [Group G] {N : Type u_5} [Group N] {ι : Sort u_7} (f : G →* N) (s : ιSubgroup N) :
                                  ⨆ (i : ι), Subgroup.comap f (s i) Subgroup.comap f (iSup s)
                                  theorem AddSubgroup.iSup_comap_le {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {ι : Sort u_7} (f : G →+ N) (s : ιAddSubgroup N) :
                                  ⨆ (i : ι), AddSubgroup.comap f (s i) AddSubgroup.comap f (iSup s)
                                  theorem Subgroup.comap_inf {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup N) (f : G →* N) :
                                  theorem AddSubgroup.comap_inf {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup N) (f : G →+ N) :
                                  theorem Subgroup.comap_iInf {G : Type u_1} [Group G] {N : Type u_5} [Group N] {ι : Sort u_7} (f : G →* N) (s : ιSubgroup N) :
                                  Subgroup.comap f (iInf s) = ⨅ (i : ι), Subgroup.comap f (s i)
                                  theorem AddSubgroup.comap_iInf {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {ι : Sort u_7} (f : G →+ N) (s : ιAddSubgroup N) :
                                  AddSubgroup.comap f (iInf s) = ⨅ (i : ι), AddSubgroup.comap f (s i)
                                  theorem Subgroup.map_inf_le {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup G) (f : G →* N) :
                                  theorem AddSubgroup.map_inf_le {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N) :
                                  theorem Subgroup.map_inf_eq {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H K : Subgroup G) (f : G →* N) (hf : Function.Injective f) :
                                  theorem AddSubgroup.map_inf_eq {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N) (hf : Function.Injective f) :
                                  @[simp]
                                  theorem Subgroup.map_bot {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                  @[simp]
                                  theorem AddSubgroup.map_bot {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                  @[simp]
                                  theorem Subgroup.map_top_of_surjective {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (h : Function.Surjective f) :
                                  @[simp]
                                  theorem AddSubgroup.map_top_of_surjective {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (h : Function.Surjective f) :
                                  @[simp]
                                  theorem Subgroup.comap_top {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                  @[simp]
                                  theorem AddSubgroup.comap_top {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                  def Subgroup.subgroupOf {G : Type u_1} [Group G] (H K : Subgroup G) :

                                  For any subgroups H and K, view H ⊓ K as a subgroup of K.

                                  Equations
                                  Instances For

                                    For any subgroups H and K, view H ⊓ K as a subgroup of K.

                                    Equations
                                    Instances For
                                      def Subgroup.subgroupOfEquivOfLe {G : Type u_7} [Group G] {H K : Subgroup G} (h : H K) :
                                      (H.subgroupOf K) ≃* H

                                      If H ≤ K, then H as a subgroup of K is isomorphic to H.

                                      Equations
                                      • Subgroup.subgroupOfEquivOfLe h = { toFun := fun (g : (H.subgroupOf K)) => g, , invFun := fun (g : H) => g, , , left_inv := , right_inv := , map_mul' := }
                                      Instances For
                                        def AddSubgroup.addSubgroupOfEquivOfLe {G : Type u_7} [AddGroup G] {H K : AddSubgroup G} (h : H K) :
                                        (H.addSubgroupOf K) ≃+ H

                                        If H ≤ K, then H as a subgroup of K is isomorphic to H.

                                        Equations
                                        • AddSubgroup.addSubgroupOfEquivOfLe h = { toFun := fun (g : (H.addSubgroupOf K)) => g, , invFun := fun (g : H) => g, , , left_inv := , right_inv := , map_add' := }
                                        Instances For
                                          @[simp]
                                          theorem AddSubgroup.addSubgroupOfEquivOfLe_symm_apply_coe_coe {G : Type u_7} [AddGroup G] {H K : AddSubgroup G} (h : H K) (g : H) :
                                          ((AddSubgroup.addSubgroupOfEquivOfLe h).symm g) = g
                                          @[simp]
                                          theorem Subgroup.subgroupOfEquivOfLe_symm_apply_coe_coe {G : Type u_7} [Group G] {H K : Subgroup G} (h : H K) (g : H) :
                                          ((Subgroup.subgroupOfEquivOfLe h).symm g) = g
                                          @[simp]
                                          theorem Subgroup.subgroupOfEquivOfLe_apply_coe {G : Type u_7} [Group G] {H K : Subgroup G} (h : H K) (g : (H.subgroupOf K)) :
                                          ((Subgroup.subgroupOfEquivOfLe h) g) = g
                                          @[simp]
                                          theorem AddSubgroup.addSubgroupOfEquivOfLe_apply_coe {G : Type u_7} [AddGroup G] {H K : AddSubgroup G} (h : H K) (g : (H.addSubgroupOf K)) :
                                          @[simp]
                                          theorem Subgroup.comap_subtype {G : Type u_1} [Group G] (H K : Subgroup G) :
                                          Subgroup.comap K.subtype H = H.subgroupOf K
                                          @[simp]
                                          theorem AddSubgroup.comap_subtype {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) :
                                          AddSubgroup.comap K.subtype H = H.addSubgroupOf K
                                          @[simp]
                                          theorem Subgroup.comap_inclusion_subgroupOf {G : Type u_1} [Group G] {K₁ K₂ : Subgroup G} (h : K₁ K₂) (H : Subgroup G) :
                                          Subgroup.comap (Subgroup.inclusion h) (H.subgroupOf K₂) = H.subgroupOf K₁
                                          @[simp]
                                          theorem AddSubgroup.comap_inclusion_addSubgroupOf {G : Type u_1} [AddGroup G] {K₁ K₂ : AddSubgroup G} (h : K₁ K₂) (H : AddSubgroup G) :
                                          AddSubgroup.comap (AddSubgroup.inclusion h) (H.addSubgroupOf K₂) = H.addSubgroupOf K₁
                                          theorem Subgroup.coe_subgroupOf {G : Type u_1} [Group G] (H K : Subgroup G) :
                                          (H.subgroupOf K) = K.subtype ⁻¹' H
                                          theorem AddSubgroup.coe_addSubgroupOf {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) :
                                          (H.addSubgroupOf K) = K.subtype ⁻¹' H
                                          theorem Subgroup.mem_subgroupOf {G : Type u_1} [Group G] {H K : Subgroup G} {h : K} :
                                          h H.subgroupOf K h H
                                          theorem AddSubgroup.mem_addSubgroupOf {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} {h : K} :
                                          h H.addSubgroupOf K h H
                                          @[simp]
                                          theorem Subgroup.subgroupOf_map_subtype {G : Type u_1} [Group G] (H K : Subgroup G) :
                                          Subgroup.map K.subtype (H.subgroupOf K) = H K
                                          @[simp]
                                          theorem AddSubgroup.addSubgroupOf_map_subtype {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) :
                                          AddSubgroup.map K.subtype (H.addSubgroupOf K) = H K
                                          @[simp]
                                          theorem Subgroup.bot_subgroupOf {G : Type u_1} [Group G] (H : Subgroup G) :
                                          .subgroupOf H =
                                          @[simp]
                                          theorem AddSubgroup.bot_addSubgroupOf {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
                                          .addSubgroupOf H =
                                          @[simp]
                                          theorem Subgroup.top_subgroupOf {G : Type u_1} [Group G] (H : Subgroup G) :
                                          .subgroupOf H =
                                          @[simp]
                                          theorem AddSubgroup.top_addSubgroupOf {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
                                          .addSubgroupOf H =
                                          theorem Subgroup.subgroupOf_bot_eq_bot {G : Type u_1} [Group G] (H : Subgroup G) :
                                          H.subgroupOf =
                                          theorem AddSubgroup.addSubgroupOf_bot_eq_bot {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
                                          H.addSubgroupOf =
                                          theorem Subgroup.subgroupOf_bot_eq_top {G : Type u_1} [Group G] (H : Subgroup G) :
                                          H.subgroupOf =
                                          theorem AddSubgroup.addSubgroupOf_bot_eq_top {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
                                          H.addSubgroupOf =
                                          @[simp]
                                          theorem Subgroup.subgroupOf_self {G : Type u_1} [Group G] (H : Subgroup G) :
                                          H.subgroupOf H =
                                          @[simp]
                                          theorem AddSubgroup.addSubgroupOf_self {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
                                          H.addSubgroupOf H =
                                          @[simp]
                                          theorem Subgroup.subgroupOf_inj {G : Type u_1} [Group G] {H₁ H₂ K : Subgroup G} :
                                          H₁.subgroupOf K = H₂.subgroupOf K H₁ K = H₂ K
                                          @[simp]
                                          theorem AddSubgroup.addSubgroupOf_inj {G : Type u_1} [AddGroup G] {H₁ H₂ K : AddSubgroup G} :
                                          H₁.addSubgroupOf K = H₂.addSubgroupOf K H₁ K = H₂ K
                                          @[simp]
                                          theorem Subgroup.inf_subgroupOf_right {G : Type u_1} [Group G] (H K : Subgroup G) :
                                          (H K).subgroupOf K = H.subgroupOf K
                                          @[simp]
                                          theorem AddSubgroup.inf_addSubgroupOf_right {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) :
                                          (H K).addSubgroupOf K = H.addSubgroupOf K
                                          @[simp]
                                          theorem Subgroup.inf_subgroupOf_left {G : Type u_1} [Group G] (H K : Subgroup G) :
                                          (K H).subgroupOf K = H.subgroupOf K
                                          @[simp]
                                          theorem AddSubgroup.inf_addSubgroupOf_left {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) :
                                          (K H).addSubgroupOf K = H.addSubgroupOf K
                                          @[simp]
                                          theorem Subgroup.subgroupOf_eq_bot {G : Type u_1} [Group G] {H K : Subgroup G} :
                                          H.subgroupOf K = Disjoint H K
                                          @[simp]
                                          theorem AddSubgroup.addSubgroupOf_eq_bot {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} :
                                          H.addSubgroupOf K = Disjoint H K
                                          @[simp]
                                          theorem Subgroup.subgroupOf_eq_top {G : Type u_1} [Group G] {H K : Subgroup G} :
                                          H.subgroupOf K = K H
                                          @[simp]
                                          theorem AddSubgroup.addSubgroupOf_eq_top {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} :
                                          H.addSubgroupOf K = K H
                                          def Subgroup.prod {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) (K : Subgroup N) :
                                          Subgroup (G × N)

                                          Given Subgroups H, K of groups G, N respectively, H × K as a subgroup of G × N.

                                          Equations
                                          • H.prod K = { toSubmonoid := H.prod K.toSubmonoid, inv_mem' := }
                                          Instances For
                                            def AddSubgroup.prod {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (K : AddSubgroup N) :

                                            Given AddSubgroups H, K of AddGroups A, B respectively, H × K as an AddSubgroup of A × B.

                                            Equations
                                            • H.prod K = { toAddSubmonoid := H.prod K.toAddSubmonoid, neg_mem' := }
                                            Instances For
                                              theorem Subgroup.coe_prod {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) (K : Subgroup N) :
                                              (H.prod K) = H ×ˢ K
                                              theorem AddSubgroup.coe_prod {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (K : AddSubgroup N) :
                                              (H.prod K) = H ×ˢ K
                                              theorem Subgroup.mem_prod {G : Type u_1} [Group G] {N : Type u_5} [Group N] {H : Subgroup G} {K : Subgroup N} {p : G × N} :
                                              p H.prod K p.1 H p.2 K
                                              theorem AddSubgroup.mem_prod {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup G} {K : AddSubgroup N} {p : G × N} :
                                              p H.prod K p.1 H p.2 K
                                              theorem Subgroup.prod_mono {G : Type u_1} [Group G] {N : Type u_5} [Group N] :
                                              ((fun (x1 x2 : Subgroup G) => x1 x2) (fun (x1 x2 : Subgroup N) => x1 x2) fun (x1 x2 : Subgroup (G × N)) => x1 x2) Subgroup.prod Subgroup.prod
                                              theorem AddSubgroup.prod_mono {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] :
                                              ((fun (x1 x2 : AddSubgroup G) => x1 x2) (fun (x1 x2 : AddSubgroup N) => x1 x2) fun (x1 x2 : AddSubgroup (G × N)) => x1 x2) AddSubgroup.prod AddSubgroup.prod
                                              theorem Subgroup.prod_mono_right {G : Type u_1} [Group G] {N : Type u_5} [Group N] (K : Subgroup G) :
                                              Monotone fun (t : Subgroup N) => K.prod t
                                              theorem AddSubgroup.prod_mono_right {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (K : AddSubgroup G) :
                                              Monotone fun (t : AddSubgroup N) => K.prod t
                                              theorem Subgroup.prod_mono_left {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup N) :
                                              Monotone fun (K : Subgroup G) => K.prod H
                                              theorem AddSubgroup.prod_mono_left {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup N) :
                                              Monotone fun (K : AddSubgroup G) => K.prod H
                                              theorem Subgroup.prod_top {G : Type u_1} [Group G] {N : Type u_5} [Group N] (K : Subgroup G) :
                                              theorem AddSubgroup.prod_top {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (K : AddSubgroup G) :
                                              theorem Subgroup.top_prod {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup N) :
                                              theorem AddSubgroup.top_prod {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup N) :
                                              @[simp]
                                              theorem Subgroup.top_prod_top {G : Type u_1} [Group G] {N : Type u_5} [Group N] :
                                              .prod =
                                              @[simp]
                                              theorem AddSubgroup.top_prod_top {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] :
                                              .prod =
                                              theorem Subgroup.bot_prod_bot {G : Type u_1} [Group G] {N : Type u_5} [Group N] :
                                              .prod =
                                              theorem AddSubgroup.bot_sum_bot {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] :
                                              .prod =
                                              theorem Subgroup.le_prod_iff {G : Type u_1} [Group G] {N : Type u_5} [Group N] {H : Subgroup G} {K : Subgroup N} {J : Subgroup (G × N)} :
                                              theorem AddSubgroup.le_prod_iff {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup G} {K : AddSubgroup N} {J : AddSubgroup (G × N)} :
                                              theorem Subgroup.prod_le_iff {G : Type u_1} [Group G] {N : Type u_5} [Group N] {H : Subgroup G} {K : Subgroup N} {J : Subgroup (G × N)} :
                                              theorem AddSubgroup.prod_le_iff {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup G} {K : AddSubgroup N} {J : AddSubgroup (G × N)} :
                                              @[simp]
                                              theorem Subgroup.prod_eq_bot_iff {G : Type u_1} [Group G] {N : Type u_5} [Group N] {H : Subgroup G} {K : Subgroup N} :
                                              H.prod K = H = K =
                                              @[simp]
                                              theorem AddSubgroup.prod_eq_bot_iff {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup G} {K : AddSubgroup N} :
                                              H.prod K = H = K =
                                              theorem Subgroup.closure_prod {G : Type u_1} [Group G] {N : Type u_5} [Group N] {s : Set G} {t : Set N} (hs : 1 s) (ht : 1 t) :
                                              theorem AddSubgroup.closure_prod {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {s : Set G} {t : Set N} (hs : 0 s) (ht : 0 t) :
                                              def Subgroup.prodEquiv {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) (K : Subgroup N) :
                                              (H.prod K) ≃* H × K

                                              Product of subgroups is isomorphic to their product as groups.

                                              Equations
                                              Instances For
                                                def AddSubgroup.prodEquiv {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (K : AddSubgroup N) :
                                                (H.prod K) ≃+ H × K

                                                Product of additive subgroups is isomorphic to their product as additive groups

                                                Equations
                                                Instances For
                                                  def Submonoid.pi {η : Type u_7} {f : ηType u_8} [(i : η) → MulOneClass (f i)] (I : Set η) (s : (i : η) → Submonoid (f i)) :
                                                  Submonoid ((i : η) → f i)

                                                  A version of Set.pi for submonoids. Given an index set I and a family of submodules s : Π i, Submonoid f i, pi I s is the submonoid of dependent functions f : Π i, f i such that f i belongs to Pi I s whenever i ∈ I.

                                                  Equations
                                                  • Submonoid.pi I s = { carrier := I.pi fun (i : η) => (s i).carrier, mul_mem' := , one_mem' := }
                                                  Instances For
                                                    def AddSubmonoid.pi {η : Type u_7} {f : ηType u_8} [(i : η) → AddZeroClass (f i)] (I : Set η) (s : (i : η) → AddSubmonoid (f i)) :
                                                    AddSubmonoid ((i : η) → f i)

                                                    A version of Set.pi for AddSubmonoids. Given an index set I and a family of submodules s : Π i, AddSubmonoid f i, pi I s is the AddSubmonoid of dependent functions f : Π i, f i such that f i belongs to pi I s whenever i ∈ I.

                                                    Equations
                                                    • AddSubmonoid.pi I s = { carrier := I.pi fun (i : η) => (s i).carrier, add_mem' := , zero_mem' := }
                                                    Instances For
                                                      def Subgroup.pi {η : Type u_7} {f : ηType u_8} [(i : η) → Group (f i)] (I : Set η) (H : (i : η) → Subgroup (f i)) :
                                                      Subgroup ((i : η) → f i)

                                                      A version of Set.pi for subgroups. Given an index set I and a family of submodules s : Π i, Subgroup f i, pi I s is the subgroup of dependent functions f : Π i, f i such that f i belongs to pi I s whenever i ∈ I.

                                                      Equations
                                                      Instances For
                                                        def AddSubgroup.pi {η : Type u_7} {f : ηType u_8} [(i : η) → AddGroup (f i)] (I : Set η) (H : (i : η) → AddSubgroup (f i)) :
                                                        AddSubgroup ((i : η) → f i)

                                                        A version of Set.pi for AddSubgroups. Given an index set I and a family of submodules s : Π i, AddSubgroup f i, pi I s is the AddSubgroup of dependent functions f : Π i, f i such that f i belongs to pi I s whenever i ∈ I.

                                                        Equations
                                                        Instances For
                                                          theorem Subgroup.coe_pi {η : Type u_7} {f : ηType u_8} [(i : η) → Group (f i)] (I : Set η) (H : (i : η) → Subgroup (f i)) :
                                                          (Subgroup.pi I H) = I.pi fun (i : η) => (H i)
                                                          theorem AddSubgroup.coe_pi {η : Type u_7} {f : ηType u_8} [(i : η) → AddGroup (f i)] (I : Set η) (H : (i : η) → AddSubgroup (f i)) :
                                                          (AddSubgroup.pi I H) = I.pi fun (i : η) => (H i)
                                                          theorem Subgroup.mem_pi {η : Type u_7} {f : ηType u_8} [(i : η) → Group (f i)] (I : Set η) {H : (i : η) → Subgroup (f i)} {p : (i : η) → f i} :
                                                          p Subgroup.pi I H iI, p i H i
                                                          theorem AddSubgroup.mem_pi {η : Type u_7} {f : ηType u_8} [(i : η) → AddGroup (f i)] (I : Set η) {H : (i : η) → AddSubgroup (f i)} {p : (i : η) → f i} :
                                                          p AddSubgroup.pi I H iI, p i H i
                                                          theorem Subgroup.pi_top {η : Type u_7} {f : ηType u_8} [(i : η) → Group (f i)] (I : Set η) :
                                                          (Subgroup.pi I fun (i : η) => ) =
                                                          theorem AddSubgroup.pi_top {η : Type u_7} {f : ηType u_8} [(i : η) → AddGroup (f i)] (I : Set η) :
                                                          (AddSubgroup.pi I fun (i : η) => ) =
                                                          theorem Subgroup.pi_empty {η : Type u_7} {f : ηType u_8} [(i : η) → Group (f i)] (H : (i : η) → Subgroup (f i)) :
                                                          theorem AddSubgroup.pi_empty {η : Type u_7} {f : ηType u_8} [(i : η) → AddGroup (f i)] (H : (i : η) → AddSubgroup (f i)) :
                                                          theorem Subgroup.pi_bot {η : Type u_7} {f : ηType u_8} [(i : η) → Group (f i)] :
                                                          (Subgroup.pi Set.univ fun (i : η) => ) =
                                                          theorem AddSubgroup.pi_bot {η : Type u_7} {f : ηType u_8} [(i : η) → AddGroup (f i)] :
                                                          (AddSubgroup.pi Set.univ fun (i : η) => ) =
                                                          theorem Subgroup.le_pi_iff {η : Type u_7} {f : ηType u_8} [(i : η) → Group (f i)] {I : Set η} {H : (i : η) → Subgroup (f i)} {J : Subgroup ((i : η) → f i)} :
                                                          J Subgroup.pi I H iI, Subgroup.map (Pi.evalMonoidHom f i) J H i
                                                          theorem AddSubgroup.le_pi_iff {η : Type u_7} {f : ηType u_8} [(i : η) → AddGroup (f i)] {I : Set η} {H : (i : η) → AddSubgroup (f i)} {J : AddSubgroup ((i : η) → f i)} :
                                                          @[simp]
                                                          theorem Subgroup.mulSingle_mem_pi {η : Type u_7} {f : ηType u_8} [(i : η) → Group (f i)] [DecidableEq η] {I : Set η} {H : (i : η) → Subgroup (f i)} (i : η) (x : f i) :
                                                          Pi.mulSingle i x Subgroup.pi I H i Ix H i
                                                          @[simp]
                                                          theorem AddSubgroup.single_mem_pi {η : Type u_7} {f : ηType u_8} [(i : η) → AddGroup (f i)] [DecidableEq η] {I : Set η} {H : (i : η) → AddSubgroup (f i)} (i : η) (x : f i) :
                                                          Pi.single i x AddSubgroup.pi I H i Ix H i
                                                          theorem Subgroup.pi_eq_bot_iff {η : Type u_7} {f : ηType u_8} [(i : η) → Group (f i)] (H : (i : η) → Subgroup (f i)) :
                                                          Subgroup.pi Set.univ H = ∀ (i : η), H i =
                                                          theorem AddSubgroup.pi_eq_bot_iff {η : Type u_7} {f : ηType u_8} [(i : η) → AddGroup (f i)] (H : (i : η) → AddSubgroup (f i)) :
                                                          AddSubgroup.pi Set.univ H = ∀ (i : η), H i =
                                                          class Subgroup.Characteristic {G : Type u_1} [Group G] (H : Subgroup G) :

                                                          A subgroup is characteristic if it is fixed by all automorphisms. Several equivalent conditions are provided by lemmas of the form Characteristic.iff...

                                                          • fixed : ∀ (ϕ : G ≃* G), Subgroup.comap ϕ.toMonoidHom H = H

                                                            H is fixed by all automorphisms

                                                          Instances
                                                            @[instance 100]
                                                            instance Subgroup.normal_of_characteristic {G : Type u_1} [Group G] (H : Subgroup G) [h : H.Characteristic] :
                                                            H.Normal
                                                            Equations
                                                            • =

                                                            An AddSubgroup is characteristic if it is fixed by all automorphisms. Several equivalent conditions are provided by lemmas of the form Characteristic.iff...

                                                            Instances
                                                              @[instance 100]
                                                              instance AddSubgroup.normal_of_characteristic {A : Type u_4} [AddGroup A] (H : AddSubgroup A) [h : H.Characteristic] :
                                                              H.Normal
                                                              Equations
                                                              • =
                                                              theorem Subgroup.characteristic_iff_comap_eq {G : Type u_1} [Group G] {H : Subgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃* G), Subgroup.comap ϕ.toMonoidHom H = H
                                                              theorem AddSubgroup.characteristic_iff_comap_eq {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃+ G), AddSubgroup.comap ϕ.toAddMonoidHom H = H
                                                              theorem Subgroup.characteristic_iff_comap_le {G : Type u_1} [Group G] {H : Subgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃* G), Subgroup.comap ϕ.toMonoidHom H H
                                                              theorem AddSubgroup.characteristic_iff_comap_le {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃+ G), AddSubgroup.comap ϕ.toAddMonoidHom H H
                                                              theorem Subgroup.characteristic_iff_le_comap {G : Type u_1} [Group G] {H : Subgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃* G), H Subgroup.comap ϕ.toMonoidHom H
                                                              theorem AddSubgroup.characteristic_iff_le_comap {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃+ G), H AddSubgroup.comap ϕ.toAddMonoidHom H
                                                              theorem Subgroup.characteristic_iff_map_eq {G : Type u_1} [Group G] {H : Subgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃* G), Subgroup.map ϕ.toMonoidHom H = H
                                                              theorem AddSubgroup.characteristic_iff_map_eq {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃+ G), AddSubgroup.map ϕ.toAddMonoidHom H = H
                                                              theorem Subgroup.characteristic_iff_map_le {G : Type u_1} [Group G] {H : Subgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃* G), Subgroup.map ϕ.toMonoidHom H H
                                                              theorem AddSubgroup.characteristic_iff_map_le {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃+ G), AddSubgroup.map ϕ.toAddMonoidHom H H
                                                              theorem Subgroup.characteristic_iff_le_map {G : Type u_1} [Group G] {H : Subgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃* G), H Subgroup.map ϕ.toMonoidHom H
                                                              theorem AddSubgroup.characteristic_iff_le_map {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
                                                              H.Characteristic ∀ (ϕ : G ≃+ G), H AddSubgroup.map ϕ.toAddMonoidHom H
                                                              instance Subgroup.botCharacteristic {G : Type u_1} [Group G] :
                                                              .Characteristic
                                                              Equations
                                                              • =
                                                              instance AddSubgroup.botCharacteristic {G : Type u_1} [AddGroup G] :
                                                              .Characteristic
                                                              Equations
                                                              • =
                                                              instance Subgroup.topCharacteristic {G : Type u_1} [Group G] :
                                                              .Characteristic
                                                              Equations
                                                              • =
                                                              instance AddSubgroup.topCharacteristic {G : Type u_1} [AddGroup G] :
                                                              .Characteristic
                                                              Equations
                                                              • =
                                                              @[instance 100]
                                                              instance Subgroup.normal_in_normalizer {G : Type u_1} [Group G] {H : Subgroup G} :
                                                              (H.subgroupOf H.normalizer).Normal
                                                              Equations
                                                              • =
                                                              @[instance 100]
                                                              instance AddSubgroup.normal_in_normalizer {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
                                                              (H.addSubgroupOf H.normalizer).Normal
                                                              Equations
                                                              • =
                                                              theorem Subgroup.normalizer_eq_top {G : Type u_1} [Group G] {H : Subgroup G} :
                                                              H.normalizer = H.Normal
                                                              theorem AddSubgroup.normalizer_eq_top {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
                                                              H.normalizer = H.Normal
                                                              theorem Subgroup.le_normalizer_of_normal {G : Type u_1} [Group G] {H K : Subgroup G} [hK : (H.subgroupOf K).Normal] (HK : H K) :
                                                              K H.normalizer
                                                              theorem AddSubgroup.le_normalizer_of_normal {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} [hK : (H.addSubgroupOf K).Normal] (HK : H K) :
                                                              K H.normalizer
                                                              theorem Subgroup.le_normalizer_comap {G : Type u_1} [Group G] {H : Subgroup G} {N : Type u_5} [Group N] (f : N →* G) :
                                                              Subgroup.comap f H.normalizer (Subgroup.comap f H).normalizer

                                                              The preimage of the normalizer is contained in the normalizer of the preimage.

                                                              theorem AddSubgroup.le_normalizer_comap {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {N : Type u_5} [AddGroup N] (f : N →+ G) :
                                                              AddSubgroup.comap f H.normalizer (AddSubgroup.comap f H).normalizer

                                                              The preimage of the normalizer is contained in the normalizer of the preimage.

                                                              theorem Subgroup.le_normalizer_map {G : Type u_1} [Group G] {H : Subgroup G} {N : Type u_5} [Group N] (f : G →* N) :
                                                              Subgroup.map f H.normalizer (Subgroup.map f H).normalizer

                                                              The image of the normalizer is contained in the normalizer of the image.

                                                              theorem AddSubgroup.le_normalizer_map {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                              AddSubgroup.map f H.normalizer (AddSubgroup.map f H).normalizer

                                                              The image of the normalizer is contained in the normalizer of the image.

                                                              def NormalizerCondition (G : Type u_1) [Group G] :

                                                              Every proper subgroup H of G is a proper normal subgroup of the normalizer of H in G.

                                                              Equations
                                                              Instances For
                                                                theorem normalizerCondition_iff_only_full_group_self_normalizing {G : Type u_1} [Group G] :
                                                                NormalizerCondition G ∀ (H : Subgroup G), H.normalizer = HH =

                                                                Alternative phrasing of the normalizer condition: Only the full group is self-normalizing. This may be easier to work with, as it avoids inequalities and negations.

                                                                instance Subgroup.map_isCommutative {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) (f : G →* G') [H.IsCommutative] :
                                                                (Subgroup.map f H).IsCommutative
                                                                Equations
                                                                • =
                                                                instance AddSubgroup.map_isCommutative {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G →+ G') [H.IsCommutative] :
                                                                (AddSubgroup.map f H).IsCommutative
                                                                Equations
                                                                • =
                                                                theorem Subgroup.comap_injective_isCommutative {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) {f : G' →* G} (hf : Function.Injective f) [H.IsCommutative] :
                                                                (Subgroup.comap f H).IsCommutative
                                                                theorem AddSubgroup.comap_injective_isCommutative {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G' →+ G} (hf : Function.Injective f) [H.IsCommutative] :
                                                                (AddSubgroup.comap f H).IsCommutative
                                                                instance Subgroup.subgroupOf_isCommutative {G : Type u_1} [Group G] (H : Subgroup G) {K : Subgroup G} [H.IsCommutative] :
                                                                (H.subgroupOf K).IsCommutative
                                                                Equations
                                                                • =
                                                                instance AddSubgroup.addSubgroupOf_isCommutative {G : Type u_1} [AddGroup G] (H : AddSubgroup G) {K : AddSubgroup G} [H.IsCommutative] :
                                                                (H.addSubgroupOf K).IsCommutative
                                                                Equations
                                                                • =
                                                                def MulEquiv.comapSubgroup {G : Type u_1} [Group G] {H : Type u_5} [Group H] (f : G ≃* H) :

                                                                An isomorphism of groups gives an order isomorphism between the lattices of subgroups, defined by sending subgroups to their inverse images.

                                                                See also MulEquiv.mapSubgroup which maps subgroups to their forward images.

                                                                Equations
                                                                Instances For
                                                                  @[simp]
                                                                  theorem MulEquiv.comapSubgroup_symm_apply {G : Type u_1} [Group G] {H : Type u_5} [Group H] (f : G ≃* H) (H✝ : Subgroup G) :
                                                                  (RelIso.symm f.comapSubgroup) H✝ = Subgroup.comap (↑f.symm) H✝
                                                                  @[simp]
                                                                  theorem MulEquiv.comapSubgroup_apply {G : Type u_1} [Group G] {H : Type u_5} [Group H] (f : G ≃* H) (H✝ : Subgroup H) :
                                                                  f.comapSubgroup H✝ = Subgroup.comap (↑f) H✝
                                                                  def MulEquiv.mapSubgroup {G : Type u_1} [Group G] {H : Type u_6} [Group H] (f : G ≃* H) :

                                                                  An isomorphism of groups gives an order isomorphism between the lattices of subgroups, defined by sending subgroups to their forward images.

                                                                  See also MulEquiv.comapSubgroup which maps subgroups to their inverse images.

                                                                  Equations
                                                                  • f.mapSubgroup = { toFun := Subgroup.map f, invFun := Subgroup.map f.symm, left_inv := , right_inv := , map_rel_iff' := }
                                                                  Instances For
                                                                    @[simp]
                                                                    theorem MulEquiv.mapSubgroup_apply {G : Type u_1} [Group G] {H : Type u_6} [Group H] (f : G ≃* H) (H✝ : Subgroup G) :
                                                                    f.mapSubgroup H✝ = Subgroup.map (↑f) H✝
                                                                    @[simp]
                                                                    theorem MulEquiv.mapSubgroup_symm_apply {G : Type u_1} [Group G] {H : Type u_6} [Group H] (f : G ≃* H) (H✝ : Subgroup H) :
                                                                    (RelIso.symm f.mapSubgroup) H✝ = Subgroup.map (↑f.symm) H✝
                                                                    def Group.conjugatesOfSet {G : Type u_1} [Group G] (s : Set G) :
                                                                    Set G

                                                                    Given a set s, conjugatesOfSet s is the set of all conjugates of the elements of s.

                                                                    Equations
                                                                    Instances For
                                                                      theorem Group.mem_conjugatesOfSet_iff {G : Type u_1} [Group G] {s : Set G} {x : G} :
                                                                      x Group.conjugatesOfSet s as, IsConj a x
                                                                      theorem Group.conjugates_subset_normal {G : Type u_1} [Group G] {N : Subgroup G} [tn : N.Normal] {a : G} (h : a N) :
                                                                      theorem Group.conjugatesOfSet_subset {G : Type u_1} [Group G] {s : Set G} {N : Subgroup G} [N.Normal] (h : s N) :
                                                                      theorem Group.conj_mem_conjugatesOfSet {G : Type u_1} [Group G] {s : Set G} {x c : G} :

                                                                      The set of conjugates of s is closed under conjugation.

                                                                      def Subgroup.normalClosure {G : Type u_1} [Group G] (s : Set G) :

                                                                      The normal closure of a set s is the subgroup closure of all the conjugates of elements of s. It is the smallest normal subgroup containing s.

                                                                      Equations
                                                                      Instances For
                                                                        instance Subgroup.normalClosure_normal {G : Type u_1} [Group G] {s : Set G} :

                                                                        The normal closure of s is a normal subgroup.

                                                                        Equations
                                                                        • =
                                                                        theorem Subgroup.normalClosure_le_normal {G : Type u_1} [Group G] {s : Set G} {N : Subgroup G} [N.Normal] (h : s N) :

                                                                        The normal closure of s is the smallest normal subgroup containing s.

                                                                        theorem Subgroup.normalClosure_subset_iff {G : Type u_1} [Group G] {s : Set G} {N : Subgroup G} [N.Normal] :
                                                                        theorem Subgroup.normalClosure_eq_iInf {G : Type u_1} [Group G] {s : Set G} :
                                                                        Subgroup.normalClosure s = ⨅ (N : Subgroup G), ⨅ (_ : N.Normal), ⨅ (_ : s N), N
                                                                        @[simp]
                                                                        theorem Subgroup.normalClosure_eq_self {G : Type u_1} [Group G] (H : Subgroup G) [H.Normal] :
                                                                        def Subgroup.normalCore {G : Type u_1} [Group G] (H : Subgroup G) :

                                                                        The normal core of a subgroup H is the largest normal subgroup of G contained in H, as shown by Subgroup.normalCore_eq_iSup.

                                                                        Equations
                                                                        • H.normalCore = { carrier := {a : G | ∀ (b : G), b * a * b⁻¹ H}, mul_mem' := , one_mem' := , inv_mem' := }
                                                                        Instances For
                                                                          theorem Subgroup.normalCore_le {G : Type u_1} [Group G] (H : Subgroup G) :
                                                                          H.normalCore H
                                                                          instance Subgroup.normalCore_normal {G : Type u_1} [Group G] (H : Subgroup G) :
                                                                          H.normalCore.Normal
                                                                          Equations
                                                                          • =
                                                                          theorem Subgroup.normal_le_normalCore {G : Type u_1} [Group G] {H N : Subgroup G} [hN : N.Normal] :
                                                                          N H.normalCore N H
                                                                          theorem Subgroup.normalCore_mono {G : Type u_1} [Group G] {H K : Subgroup G} (h : H K) :
                                                                          H.normalCore K.normalCore
                                                                          theorem Subgroup.normalCore_eq_iSup {G : Type u_1} [Group G] (H : Subgroup G) :
                                                                          H.normalCore = ⨆ (N : Subgroup G), ⨆ (_ : N.Normal), ⨆ (_ : N H), N
                                                                          @[simp]
                                                                          theorem Subgroup.normalCore_eq_self {G : Type u_1} [Group G] (H : Subgroup G) [H.Normal] :
                                                                          H.normalCore = H
                                                                          theorem Subgroup.normalCore_idempotent {G : Type u_1} [Group G] (H : Subgroup G) :
                                                                          H.normalCore.normalCore = H.normalCore
                                                                          def MonoidHom.range {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :

                                                                          The range of a monoid homomorphism from a group is a subgroup.

                                                                          Equations
                                                                          Instances For
                                                                            def AddMonoidHom.range {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :

                                                                            The range of an AddMonoidHom from an AddGroup is an AddSubgroup.

                                                                            Equations
                                                                            Instances For
                                                                              @[simp]
                                                                              theorem MonoidHom.coe_range {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                                                              f.range = Set.range f
                                                                              @[simp]
                                                                              theorem AddMonoidHom.coe_range {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                                              f.range = Set.range f
                                                                              @[simp]
                                                                              theorem MonoidHom.mem_range {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {y : N} :
                                                                              y f.range ∃ (x : G), f x = y
                                                                              @[simp]
                                                                              theorem AddMonoidHom.mem_range {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {y : N} :
                                                                              y f.range ∃ (x : G), f x = y
                                                                              theorem MonoidHom.range_eq_map {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                                                              f.range = Subgroup.map f
                                                                              theorem AddMonoidHom.range_eq_map {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                                              instance MonoidHom.range_isCommutative {G : Type u_7} [CommGroup G] {N : Type u_8} [Group N] (f : G →* N) :
                                                                              f.range.IsCommutative
                                                                              Equations
                                                                              • =
                                                                              instance AddMonoidHom.range_isCommutative {G : Type u_7} [AddCommGroup G] {N : Type u_8} [AddGroup N] (f : G →+ N) :
                                                                              f.range.IsCommutative
                                                                              Equations
                                                                              • =
                                                                              @[simp]
                                                                              theorem MonoidHom.restrict_range {G : Type u_1} [Group G] {N : Type u_5} [Group N] (K : Subgroup G) (f : G →* N) :
                                                                              (f.restrict K).range = Subgroup.map f K
                                                                              @[simp]
                                                                              theorem AddMonoidHom.restrict_range {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (K : AddSubgroup G) (f : G →+ N) :
                                                                              (f.restrict K).range = AddSubgroup.map f K
                                                                              def MonoidHom.rangeRestrict {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                                                              G →* f.range

                                                                              The canonical surjective group homomorphism G →* f(G) induced by a group homomorphism G →* N.

                                                                              Equations
                                                                              • f.rangeRestrict = f.codRestrict f.range
                                                                              Instances For
                                                                                def AddMonoidHom.rangeRestrict {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                                                G →+ f.range

                                                                                The canonical surjective AddGroup homomorphism G →+ f(G) induced by a group homomorphism G →+ N.

                                                                                Equations
                                                                                • f.rangeRestrict = f.codRestrict f.range
                                                                                Instances For
                                                                                  @[simp]
                                                                                  theorem MonoidHom.coe_rangeRestrict {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (g : G) :
                                                                                  (f.rangeRestrict g) = f g
                                                                                  @[simp]
                                                                                  theorem AddMonoidHom.coe_rangeRestrict {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (g : G) :
                                                                                  (f.rangeRestrict g) = f g
                                                                                  theorem MonoidHom.coe_comp_rangeRestrict {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                                                                  Subtype.val f.rangeRestrict = f
                                                                                  theorem AddMonoidHom.coe_comp_rangeRestrict {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                                                  Subtype.val f.rangeRestrict = f
                                                                                  theorem MonoidHom.subtype_comp_rangeRestrict {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                                                                  f.range.subtype.comp f.rangeRestrict = f
                                                                                  theorem AddMonoidHom.subtype_comp_rangeRestrict {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                                                  f.range.subtype.comp f.rangeRestrict = f
                                                                                  theorem MonoidHom.rangeRestrict_surjective {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                                                                  Function.Surjective f.rangeRestrict
                                                                                  theorem AddMonoidHom.rangeRestrict_surjective {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                                                  Function.Surjective f.rangeRestrict
                                                                                  @[simp]
                                                                                  theorem MonoidHom.rangeRestrict_injective_iff {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} :
                                                                                  Function.Injective f.rangeRestrict Function.Injective f
                                                                                  @[simp]
                                                                                  theorem AddMonoidHom.rangeRestrict_injective_iff {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} :
                                                                                  Function.Injective f.rangeRestrict Function.Injective f
                                                                                  theorem MonoidHom.map_range {G : Type u_1} [Group G] {N : Type u_5} {P : Type u_6} [Group N] [Group P] (g : N →* P) (f : G →* N) :
                                                                                  Subgroup.map g f.range = (g.comp f).range
                                                                                  theorem AddMonoidHom.map_range {G : Type u_1} [AddGroup G] {N : Type u_5} {P : Type u_6} [AddGroup N] [AddGroup P] (g : N →+ P) (f : G →+ N) :
                                                                                  AddSubgroup.map g f.range = (g.comp f).range
                                                                                  theorem MonoidHom.range_top_iff_surjective {G : Type u_1} [Group G] {N : Type u_7} [Group N] {f : G →* N} :
                                                                                  theorem AddMonoidHom.range_top_iff_surjective {G : Type u_1} [AddGroup G] {N : Type u_7} [AddGroup N] {f : G →+ N} :
                                                                                  @[simp]
                                                                                  theorem MonoidHom.range_top_of_surjective {G : Type u_1} [Group G] {N : Type u_7} [Group N] (f : G →* N) (hf : Function.Surjective f) :
                                                                                  f.range =

                                                                                  The range of a surjective monoid homomorphism is the whole of the codomain.

                                                                                  @[simp]
                                                                                  theorem AddMonoidHom.range_top_of_surjective {G : Type u_1} [AddGroup G] {N : Type u_7} [AddGroup N] (f : G →+ N) (hf : Function.Surjective f) :
                                                                                  f.range =

                                                                                  The range of a surjective AddMonoid homomorphism is the whole of the codomain.

                                                                                  @[simp]
                                                                                  theorem MonoidHom.range_one {G : Type u_1} [Group G] {N : Type u_5} [Group N] :
                                                                                  @[simp]
                                                                                  theorem AddMonoidHom.range_zero {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] :
                                                                                  @[simp]
                                                                                  theorem Subgroup.subtype_range {G : Type u_1} [Group G] (H : Subgroup G) :
                                                                                  H.subtype.range = H
                                                                                  @[simp]
                                                                                  theorem AddSubgroup.subtype_range {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
                                                                                  H.subtype.range = H
                                                                                  @[simp]
                                                                                  theorem Subgroup.inclusion_range {G : Type u_1} [Group G] {H K : Subgroup G} (h_le : H K) :
                                                                                  (Subgroup.inclusion h_le).range = H.subgroupOf K
                                                                                  @[simp]
                                                                                  theorem AddSubgroup.inclusion_range {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h_le : H K) :
                                                                                  (AddSubgroup.inclusion h_le).range = H.addSubgroupOf K
                                                                                  theorem MonoidHom.subgroupOf_range_eq_of_le {G₁ : Type u_7} {G₂ : Type u_8} [Group G₁] [Group G₂] {K : Subgroup G₂} (f : G₁ →* G₂) (h : f.range K) :
                                                                                  f.range.subgroupOf K = (f.codRestrict K ).range
                                                                                  theorem AddMonoidHom.addSubgroupOf_range_eq_of_le {G₁ : Type u_7} {G₂ : Type u_8} [AddGroup G₁] [AddGroup G₂] {K : AddSubgroup G₂} (f : G₁ →+ G₂) (h : f.range K) :
                                                                                  f.range.addSubgroupOf K = (f.codRestrict K ).range
                                                                                  @[simp]
                                                                                  theorem MonoidHom.coe_toAdditive_range {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') :
                                                                                  (MonoidHom.toAdditive f).range = Subgroup.toAddSubgroup f.range
                                                                                  @[simp]
                                                                                  theorem MonoidHom.coe_toMultiplicative_range {A : Type u_7} {A' : Type u_8} [AddGroup A] [AddGroup A'] (f : A →+ A') :
                                                                                  (AddMonoidHom.toMultiplicative f).range = AddSubgroup.toSubgroup f.range
                                                                                  def MonoidHom.ofLeftInverse {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {g : N →* G} (h : Function.LeftInverse g f) :
                                                                                  G ≃* f.range

                                                                                  Computable alternative to MonoidHom.ofInjective.

                                                                                  Equations
                                                                                  • MonoidHom.ofLeftInverse h = { toFun := f.rangeRestrict, invFun := g f.range.subtype, left_inv := h, right_inv := , map_mul' := }
                                                                                  Instances For
                                                                                    def AddMonoidHom.ofLeftInverse {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {g : N →+ G} (h : Function.LeftInverse g f) :
                                                                                    G ≃+ f.range

                                                                                    Computable alternative to AddMonoidHom.ofInjective.

                                                                                    Equations
                                                                                    • AddMonoidHom.ofLeftInverse h = { toFun := f.rangeRestrict, invFun := g f.range.subtype, left_inv := h, right_inv := , map_add' := }
                                                                                    Instances For
                                                                                      @[simp]
                                                                                      theorem MonoidHom.ofLeftInverse_apply {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {g : N →* G} (h : Function.LeftInverse g f) (x : G) :
                                                                                      ((MonoidHom.ofLeftInverse h) x) = f x
                                                                                      @[simp]
                                                                                      theorem AddMonoidHom.ofLeftInverse_apply {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {g : N →+ G} (h : Function.LeftInverse g f) (x : G) :
                                                                                      @[simp]
                                                                                      theorem MonoidHom.ofLeftInverse_symm_apply {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {g : N →* G} (h : Function.LeftInverse g f) (x : f.range) :
                                                                                      (MonoidHom.ofLeftInverse h).symm x = g x
                                                                                      @[simp]
                                                                                      theorem AddMonoidHom.ofLeftInverse_symm_apply {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {g : N →+ G} (h : Function.LeftInverse g f) (x : f.range) :
                                                                                      (AddMonoidHom.ofLeftInverse h).symm x = g x
                                                                                      noncomputable def MonoidHom.ofInjective {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (hf : Function.Injective f) :
                                                                                      G ≃* f.range

                                                                                      The range of an injective group homomorphism is isomorphic to its domain.

                                                                                      Equations
                                                                                      Instances For
                                                                                        noncomputable def AddMonoidHom.ofInjective {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (hf : Function.Injective f) :
                                                                                        G ≃+ f.range

                                                                                        The range of an injective additive group homomorphism is isomorphic to its domain.

                                                                                        Equations
                                                                                        Instances For
                                                                                          theorem MonoidHom.ofInjective_apply {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (hf : Function.Injective f) {x : G} :
                                                                                          ((MonoidHom.ofInjective hf) x) = f x
                                                                                          theorem AddMonoidHom.ofInjective_apply {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (hf : Function.Injective f) {x : G} :
                                                                                          ((AddMonoidHom.ofInjective hf) x) = f x
                                                                                          @[simp]
                                                                                          theorem MonoidHom.apply_ofInjective_symm {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (hf : Function.Injective f) (x : f.range) :
                                                                                          f ((MonoidHom.ofInjective hf).symm x) = x
                                                                                          @[simp]
                                                                                          theorem AddMonoidHom.apply_ofInjective_symm {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (hf : Function.Injective f) (x : f.range) :
                                                                                          f ((AddMonoidHom.ofInjective hf).symm x) = x
                                                                                          def MonoidHom.ker {G : Type u_1} [Group G] {M : Type u_7} [MulOneClass M] (f : G →* M) :

                                                                                          The multiplicative kernel of a monoid homomorphism is the subgroup of elements x : G such that f x = 1

                                                                                          Equations
                                                                                          Instances For
                                                                                            def AddMonoidHom.ker {G : Type u_1} [AddGroup G] {M : Type u_7} [AddZeroClass M] (f : G →+ M) :

                                                                                            The additive kernel of an AddMonoid homomorphism is the AddSubgroup of elements such that f x = 0

                                                                                            Equations
                                                                                            Instances For
                                                                                              @[simp]
                                                                                              theorem MonoidHom.mem_ker {G : Type u_1} [Group G] {M : Type u_7} [MulOneClass M] {f : G →* M} {x : G} :
                                                                                              x f.ker f x = 1
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.mem_ker {G : Type u_1} [AddGroup G] {M : Type u_7} [AddZeroClass M] {f : G →+ M} {x : G} :
                                                                                              x f.ker f x = 0
                                                                                              theorem MonoidHom.div_mem_ker_iff {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) {x y : G} :
                                                                                              x / y f.ker f x = f y
                                                                                              theorem AddMonoidHom.sub_mem_ker_iff {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) {x y : G} :
                                                                                              x - y f.ker f x = f y
                                                                                              theorem MonoidHom.coe_ker {G : Type u_1} [Group G] {M : Type u_7} [MulOneClass M] (f : G →* M) :
                                                                                              f.ker = f ⁻¹' {1}
                                                                                              theorem AddMonoidHom.coe_ker {G : Type u_1} [AddGroup G] {M : Type u_7} [AddZeroClass M] (f : G →+ M) :
                                                                                              f.ker = f ⁻¹' {0}
                                                                                              @[simp]
                                                                                              theorem MonoidHom.ker_toHomUnits {G : Type u_1} [Group G] {M : Type u_8} [Monoid M] (f : G →* M) :
                                                                                              f.toHomUnits.ker = f.ker
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.ker_toHomAddUnits {G : Type u_1} [AddGroup G] {M : Type u_8} [AddMonoid M] (f : G →+ M) :
                                                                                              f.toHomAddUnits.ker = f.ker
                                                                                              theorem MonoidHom.eq_iff {G : Type u_1} [Group G] {M : Type u_7} [MulOneClass M] (f : G →* M) {x y : G} :
                                                                                              f x = f y y⁻¹ * x f.ker
                                                                                              theorem AddMonoidHom.eq_iff {G : Type u_1} [AddGroup G] {M : Type u_7} [AddZeroClass M] (f : G →+ M) {x y : G} :
                                                                                              f x = f y -y + x f.ker
                                                                                              instance MonoidHom.decidableMemKer {G : Type u_1} [Group G] {M : Type u_7} [MulOneClass M] [DecidableEq M] (f : G →* M) :
                                                                                              DecidablePred fun (x : G) => x f.ker
                                                                                              Equations
                                                                                              instance AddMonoidHom.decidableMemKer {G : Type u_1} [AddGroup G] {M : Type u_7} [AddZeroClass M] [DecidableEq M] (f : G →+ M) :
                                                                                              DecidablePred fun (x : G) => x f.ker
                                                                                              Equations
                                                                                              theorem MonoidHom.comap_ker {G : Type u_1} [Group G] {N : Type u_5} {P : Type u_6} [Group N] [Group P] (g : N →* P) (f : G →* N) :
                                                                                              Subgroup.comap f g.ker = (g.comp f).ker
                                                                                              theorem AddMonoidHom.comap_ker {G : Type u_1} [AddGroup G] {N : Type u_5} {P : Type u_6} [AddGroup N] [AddGroup P] (g : N →+ P) (f : G →+ N) :
                                                                                              AddSubgroup.comap f g.ker = (g.comp f).ker
                                                                                              @[simp]
                                                                                              theorem MonoidHom.comap_bot {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.comap_bot {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                                                              @[simp]
                                                                                              theorem MonoidHom.ker_restrict {G : Type u_1} [Group G] {N : Type u_5} [Group N] (K : Subgroup G) (f : G →* N) :
                                                                                              (f.restrict K).ker = f.ker.subgroupOf K
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.ker_restrict {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (K : AddSubgroup G) (f : G →+ N) :
                                                                                              (f.restrict K).ker = f.ker.addSubgroupOf K
                                                                                              @[simp]
                                                                                              theorem MonoidHom.ker_codRestrict {G : Type u_1} [Group G] {N : Type u_5} [Group N] {S : Type u_8} [SetLike S N] [SubmonoidClass S N] (f : G →* N) (s : S) (h : ∀ (x : G), f x s) :
                                                                                              (f.codRestrict s h).ker = f.ker
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.ker_codRestrict {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {S : Type u_8} [SetLike S N] [AddSubmonoidClass S N] (f : G →+ N) (s : S) (h : ∀ (x : G), f x s) :
                                                                                              (f.codRestrict s h).ker = f.ker
                                                                                              @[simp]
                                                                                              theorem MonoidHom.ker_rangeRestrict {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                                                                              f.rangeRestrict.ker = f.ker
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.ker_rangeRestrict {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                                                              f.rangeRestrict.ker = f.ker
                                                                                              @[simp]
                                                                                              theorem MonoidHom.ker_one {G : Type u_1} [Group G] {M : Type u_7} [MulOneClass M] :
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.ker_zero {G : Type u_1} [AddGroup G] {M : Type u_7} [AddZeroClass M] :
                                                                                              @[simp]
                                                                                              theorem MonoidHom.ker_id {G : Type u_1} [Group G] :
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.ker_id {G : Type u_1} [AddGroup G] :
                                                                                              theorem MonoidHom.ker_eq_bot_iff {G : Type u_1} [Group G] {M : Type u_7} [MulOneClass M] (f : G →* M) :
                                                                                              theorem AddMonoidHom.ker_eq_bot_iff {G : Type u_1} [AddGroup G] {M : Type u_7} [AddZeroClass M] (f : G →+ M) :
                                                                                              @[simp]
                                                                                              theorem Subgroup.ker_subtype {G : Type u_1} [Group G] (H : Subgroup G) :
                                                                                              H.subtype.ker =
                                                                                              @[simp]
                                                                                              theorem AddSubgroup.ker_subtype {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
                                                                                              H.subtype.ker =
                                                                                              @[simp]
                                                                                              theorem Subgroup.ker_inclusion {G : Type u_1} [Group G] {H K : Subgroup G} (h : H K) :
                                                                                              @[simp]
                                                                                              theorem AddSubgroup.ker_inclusion {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H K) :
                                                                                              theorem MonoidHom.ker_prod {G : Type u_1} [Group G] {M : Type u_8} {N : Type u_9} [MulOneClass M] [MulOneClass N] (f : G →* M) (g : G →* N) :
                                                                                              (f.prod g).ker = f.ker g.ker
                                                                                              theorem AddMonoidHom.ker_sum {G : Type u_1} [AddGroup G] {M : Type u_8} {N : Type u_9} [AddZeroClass M] [AddZeroClass N] (f : G →+ M) (g : G →+ N) :
                                                                                              (f.prod g).ker = f.ker g.ker
                                                                                              theorem MonoidHom.prodMap_comap_prod {G : Type u_1} [Group G] {N : Type u_5} [Group N] {G' : Type u_8} {N' : Type u_9} [Group G'] [Group N'] (f : G →* N) (g : G' →* N') (S : Subgroup N) (S' : Subgroup N') :
                                                                                              Subgroup.comap (f.prodMap g) (S.prod S') = (Subgroup.comap f S).prod (Subgroup.comap g S')
                                                                                              theorem AddMonoidHom.sumMap_comap_sum {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {G' : Type u_8} {N' : Type u_9} [AddGroup G'] [AddGroup N'] (f : G →+ N) (g : G' →+ N') (S : AddSubgroup N) (S' : AddSubgroup N') :
                                                                                              AddSubgroup.comap (f.prodMap g) (S.prod S') = (AddSubgroup.comap f S).prod (AddSubgroup.comap g S')
                                                                                              theorem MonoidHom.ker_prodMap {G : Type u_1} [Group G] {N : Type u_5} [Group N] {G' : Type u_8} {N' : Type u_9} [Group G'] [Group N'] (f : G →* N) (g : G' →* N') :
                                                                                              (f.prodMap g).ker = f.ker.prod g.ker
                                                                                              theorem AddMonoidHom.ker_sumMap {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {G' : Type u_8} {N' : Type u_9} [AddGroup G'] [AddGroup N'] (f : G →+ N) (g : G' →+ N') :
                                                                                              (f.prodMap g).ker = f.ker.prod g.ker
                                                                                              theorem MonoidHom.range_le_ker_iff {G : Type u_1} {G' : Type u_2} {G'' : Type u_3} [Group G] [Group G'] [Group G''] (f : G →* G') (g : G' →* G'') :
                                                                                              f.range g.ker g.comp f = 1
                                                                                              theorem AddMonoidHom.range_le_ker_iff {G : Type u_1} {G' : Type u_2} {G'' : Type u_3} [AddGroup G] [AddGroup G'] [AddGroup G''] (f : G →+ G') (g : G' →+ G'') :
                                                                                              f.range g.ker g.comp f = 0
                                                                                              @[instance 100]
                                                                                              instance MonoidHom.normal_ker {G : Type u_1} [Group G] {M : Type u_7} [MulOneClass M] (f : G →* M) :
                                                                                              f.ker.Normal
                                                                                              Equations
                                                                                              • =
                                                                                              @[instance 100]
                                                                                              instance AddMonoidHom.normal_ker {G : Type u_1} [AddGroup G] {M : Type u_7} [AddZeroClass M] (f : G →+ M) :
                                                                                              f.ker.Normal
                                                                                              Equations
                                                                                              • =
                                                                                              @[simp]
                                                                                              theorem MonoidHom.ker_fst {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] :
                                                                                              (MonoidHom.fst G G').ker = .prod
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.ker_fst {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] :
                                                                                              (AddMonoidHom.fst G G').ker = .prod
                                                                                              @[simp]
                                                                                              theorem MonoidHom.ker_snd {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] :
                                                                                              (MonoidHom.snd G G').ker = .prod
                                                                                              @[simp]
                                                                                              theorem AddMonoidHom.ker_snd {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] :
                                                                                              (AddMonoidHom.snd G G').ker = .prod
                                                                                              @[simp]
                                                                                              theorem MonoidHom.coe_toAdditive_ker {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') :
                                                                                              (MonoidHom.toAdditive f).ker = Subgroup.toAddSubgroup f.ker
                                                                                              @[simp]
                                                                                              theorem MonoidHom.coe_toMultiplicative_ker {A : Type u_8} {A' : Type u_9} [AddGroup A] [AddGroup A'] (f : A →+ A') :
                                                                                              (AddMonoidHom.toMultiplicative f).ker = AddSubgroup.toSubgroup f.ker
                                                                                              def MonoidHom.eqLocus {G : Type u_1} [Group G] {M : Type u_7} [Monoid M] (f g : G →* M) :

                                                                                              The subgroup of elements x : G such that f x = g x

                                                                                              Equations
                                                                                              • f.eqLocus g = { toSubmonoid := f.eqLocusM g, inv_mem' := }
                                                                                              Instances For
                                                                                                def AddMonoidHom.eqLocus {G : Type u_1} [AddGroup G] {M : Type u_7} [AddMonoid M] (f g : G →+ M) :

                                                                                                The additive subgroup of elements x : G such that f x = g x

                                                                                                Equations
                                                                                                • f.eqLocus g = { toAddSubmonoid := f.eqLocusM g, neg_mem' := }
                                                                                                Instances For
                                                                                                  @[simp]
                                                                                                  theorem MonoidHom.eqLocus_same {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) :
                                                                                                  f.eqLocus f =
                                                                                                  @[simp]
                                                                                                  theorem AddMonoidHom.eqLocus_same {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) :
                                                                                                  f.eqLocus f =
                                                                                                  theorem MonoidHom.eqOn_closure {G : Type u_1} [Group G] {M : Type u_7} [Monoid M] {f g : G →* M} {s : Set G} (h : Set.EqOn (⇑f) (⇑g) s) :
                                                                                                  Set.EqOn f g (Subgroup.closure s)

                                                                                                  If two monoid homomorphisms are equal on a set, then they are equal on its subgroup closure.

                                                                                                  theorem AddMonoidHom.eqOn_closure {G : Type u_1} [AddGroup G] {M : Type u_7} [AddMonoid M] {f g : G →+ M} {s : Set G} (h : Set.EqOn (⇑f) (⇑g) s) :
                                                                                                  Set.EqOn f g (AddSubgroup.closure s)

                                                                                                  If two monoid homomorphisms are equal on a set, then they are equal on its subgroup closure.

                                                                                                  theorem MonoidHom.eq_of_eqOn_top {G : Type u_1} [Group G] {M : Type u_7} [Monoid M] {f g : G →* M} (h : Set.EqOn f g ) :
                                                                                                  f = g
                                                                                                  theorem AddMonoidHom.eq_of_eqOn_top {G : Type u_1} [AddGroup G] {M : Type u_7} [AddMonoid M] {f g : G →+ M} (h : Set.EqOn f g ) :
                                                                                                  f = g
                                                                                                  theorem MonoidHom.eq_of_eqOn_dense {G : Type u_1} [Group G] {M : Type u_7} [Monoid M] {s : Set G} (hs : Subgroup.closure s = ) {f g : G →* M} (h : Set.EqOn (⇑f) (⇑g) s) :
                                                                                                  f = g
                                                                                                  theorem AddMonoidHom.eq_of_eqOn_dense {G : Type u_1} [AddGroup G] {M : Type u_7} [AddMonoid M] {s : Set G} (hs : AddSubgroup.closure s = ) {f g : G →+ M} (h : Set.EqOn (⇑f) (⇑g) s) :
                                                                                                  f = g
                                                                                                  theorem MonoidHom.closure_preimage_le {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (s : Set N) :
                                                                                                  theorem MonoidHom.map_closure {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (s : Set G) :

                                                                                                  The image under a monoid homomorphism of the subgroup generated by a set equals the subgroup generated by the image of the set.

                                                                                                  theorem AddMonoidHom.map_closure {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (s : Set G) :

                                                                                                  The image under an AddMonoid hom of the AddSubgroup generated by a set equals the AddSubgroup generated by the image of the set.

                                                                                                  theorem Subgroup.Normal.map {G : Type u_1} [Group G] {N : Type u_5} [Group N] {H : Subgroup G} (h : H.Normal) (f : G →* N) (hf : Function.Surjective f) :
                                                                                                  (Subgroup.map f H).Normal
                                                                                                  theorem AddSubgroup.Normal.map {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup G} (h : H.Normal) (f : G →+ N) (hf : Function.Surjective f) :
                                                                                                  (AddSubgroup.map f H).Normal
                                                                                                  theorem Subgroup.map_eq_bot_iff {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) {f : G →* N} :
                                                                                                  Subgroup.map f H = H f.ker
                                                                                                  theorem AddSubgroup.map_eq_bot_iff {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) {f : G →+ N} :
                                                                                                  theorem Subgroup.map_eq_bot_iff_of_injective {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) {f : G →* N} (hf : Function.Injective f) :
                                                                                                  theorem AddSubgroup.map_eq_bot_iff_of_injective {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) {f : G →+ N} (hf : Function.Injective f) :
                                                                                                  theorem Subgroup.map_le_range {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (H : Subgroup G) :
                                                                                                  Subgroup.map f H f.range
                                                                                                  theorem AddSubgroup.map_le_range {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup G) :
                                                                                                  AddSubgroup.map f H f.range
                                                                                                  theorem Subgroup.map_subtype_le {G : Type u_1} [Group G] {H : Subgroup G} (K : Subgroup H) :
                                                                                                  Subgroup.map H.subtype K H
                                                                                                  theorem AddSubgroup.map_subtype_le {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (K : AddSubgroup H) :
                                                                                                  AddSubgroup.map H.subtype K H
                                                                                                  theorem Subgroup.ker_le_comap {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (H : Subgroup N) :
                                                                                                  theorem AddSubgroup.ker_le_comap {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup N) :
                                                                                                  theorem Subgroup.map_comap_le {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (H : Subgroup N) :
                                                                                                  theorem AddSubgroup.map_comap_le {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup N) :
                                                                                                  theorem Subgroup.le_comap_map {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (H : Subgroup G) :
                                                                                                  theorem AddSubgroup.le_comap_map {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup G) :
                                                                                                  theorem Subgroup.map_comap_eq {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (H : Subgroup N) :
                                                                                                  Subgroup.map f (Subgroup.comap f H) = f.range H
                                                                                                  theorem AddSubgroup.map_comap_eq {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup N) :
                                                                                                  theorem Subgroup.comap_map_eq {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (H : Subgroup G) :
                                                                                                  theorem AddSubgroup.comap_map_eq {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup G) :
                                                                                                  theorem Subgroup.map_comap_eq_self {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {H : Subgroup N} (h : H f.range) :
                                                                                                  theorem AddSubgroup.map_comap_eq_self {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H : AddSubgroup N} (h : H f.range) :
                                                                                                  theorem Subgroup.map_comap_eq_self_of_surjective {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (h : Function.Surjective f) (H : Subgroup N) :
                                                                                                  theorem Subgroup.comap_le_comap_of_le_range {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K L : Subgroup N} (hf : K f.range) :
                                                                                                  theorem AddSubgroup.comap_le_comap_of_le_range {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K L : AddSubgroup N} (hf : K f.range) :
                                                                                                  theorem Subgroup.comap_le_comap_of_surjective {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K L : Subgroup N} (hf : Function.Surjective f) :
                                                                                                  theorem Subgroup.comap_lt_comap_of_surjective {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {K L : Subgroup N} (hf : Function.Surjective f) :
                                                                                                  theorem Subgroup.comap_injective {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (h : Function.Surjective f) :
                                                                                                  theorem Subgroup.comap_map_eq_self {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {H : Subgroup G} (h : f.ker H) :
                                                                                                  theorem AddSubgroup.comap_map_eq_self {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H : AddSubgroup G} (h : f.ker H) :
                                                                                                  theorem Subgroup.comap_map_eq_self_of_injective {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (h : Function.Injective f) (H : Subgroup G) :
                                                                                                  theorem Subgroup.map_le_map_iff {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {H K : Subgroup G} :
                                                                                                  theorem AddSubgroup.map_le_map_iff {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H K : AddSubgroup G} :
                                                                                                  theorem Subgroup.map_le_map_iff' {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {H K : Subgroup G} :
                                                                                                  Subgroup.map f H Subgroup.map f K H f.ker K f.ker
                                                                                                  theorem AddSubgroup.map_le_map_iff' {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H K : AddSubgroup G} :
                                                                                                  AddSubgroup.map f H AddSubgroup.map f K H f.ker K f.ker
                                                                                                  theorem Subgroup.map_eq_map_iff {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {H K : Subgroup G} :
                                                                                                  Subgroup.map f H = Subgroup.map f K H f.ker = K f.ker
                                                                                                  theorem AddSubgroup.map_eq_map_iff {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H K : AddSubgroup G} :
                                                                                                  AddSubgroup.map f H = AddSubgroup.map f K H f.ker = K f.ker
                                                                                                  theorem Subgroup.map_eq_range_iff {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {H : Subgroup G} :
                                                                                                  Subgroup.map f H = f.range Codisjoint H f.ker
                                                                                                  theorem AddSubgroup.map_eq_range_iff {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H : AddSubgroup G} :
                                                                                                  AddSubgroup.map f H = f.range Codisjoint H f.ker
                                                                                                  theorem Subgroup.map_le_map_iff_of_injective {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (hf : Function.Injective f) {H K : Subgroup G} :
                                                                                                  @[simp]
                                                                                                  theorem Subgroup.map_subtype_le_map_subtype {G : Type u_1} [Group G] {G' : Subgroup G} {H K : Subgroup G'} :
                                                                                                  Subgroup.map G'.subtype H Subgroup.map G'.subtype K H K
                                                                                                  @[simp]
                                                                                                  theorem AddSubgroup.map_subtype_le_map_subtype {G : Type u_1} [AddGroup G] {G' : AddSubgroup G} {H K : AddSubgroup G'} :
                                                                                                  AddSubgroup.map G'.subtype H AddSubgroup.map G'.subtype K H K
                                                                                                  theorem Subgroup.map_injective {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} (h : Function.Injective f) :
                                                                                                  theorem Subgroup.map_eq_comap_of_inverse {G : Type u_1} [Group G] {N : Type u_5} [Group N] {f : G →* N} {g : N →* G} (hl : Function.LeftInverse g f) (hr : Function.RightInverse g f) (H : Subgroup G) :
                                                                                                  theorem AddSubgroup.map_eq_comap_of_inverse {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {g : N →+ G} (hl : Function.LeftInverse g f) (hr : Function.RightInverse g f) (H : AddSubgroup G) :
                                                                                                  theorem Subgroup.map_injective_of_ker_le {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) {H K : Subgroup G} (hH : f.ker H) (hK : f.ker K) (hf : Subgroup.map f H = Subgroup.map f K) :
                                                                                                  H = K

                                                                                                  Given f(A) = f(B), ker f ≤ A, and ker f ≤ B, deduce that A = B.

                                                                                                  theorem AddSubgroup.map_injective_of_ker_le {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) {H K : AddSubgroup G} (hH : f.ker H) (hK : f.ker K) (hf : AddSubgroup.map f H = AddSubgroup.map f K) :
                                                                                                  H = K

                                                                                                  Given f(A) = f(B), ker f ≤ A, and ker f ≤ B, deduce that A = B.

                                                                                                  theorem Subgroup.comap_sup_eq_of_le_range {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) {H K : Subgroup N} (hH : H f.range) (hK : K f.range) :
                                                                                                  theorem AddSubgroup.comap_sup_eq_of_le_range {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) {H K : AddSubgroup N} (hH : H f.range) (hK : K f.range) :
                                                                                                  theorem Subgroup.comap_sup_eq {G : Type u_1} [Group G] {N : Type u_5} [Group N] (f : G →* N) (H K : Subgroup N) (hf : Function.Surjective f) :
                                                                                                  theorem AddSubgroup.comap_sup_eq {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H K : AddSubgroup N) (hf : Function.Surjective f) :
                                                                                                  theorem Subgroup.sup_subgroupOf_eq {G : Type u_1} [Group G] {H K L : Subgroup G} (hH : H L) (hK : K L) :
                                                                                                  H.subgroupOf L K.subgroupOf L = (H K).subgroupOf L
                                                                                                  theorem AddSubgroup.sup_addSubgroupOf_eq {G : Type u_1} [AddGroup G] {H K L : AddSubgroup G} (hH : H L) (hK : K L) :
                                                                                                  H.addSubgroupOf L K.addSubgroupOf L = (H K).addSubgroupOf L
                                                                                                  theorem Subgroup.codisjoint_subgroupOf_sup {G : Type u_1} [Group G] (H K : Subgroup G) :
                                                                                                  Codisjoint (H.subgroupOf (H K)) (K.subgroupOf (H K))
                                                                                                  theorem AddSubgroup.codisjoint_addSubgroupOf_sup {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) :
                                                                                                  Codisjoint (H.addSubgroupOf (H K)) (K.addSubgroupOf (H K))
                                                                                                  noncomputable def Subgroup.equivMapOfInjective {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) (f : G →* N) (hf : Function.Injective f) :
                                                                                                  H ≃* (Subgroup.map f H)

                                                                                                  A subgroup is isomorphic to its image under an injective function. If you have an isomorphism, use MulEquiv.subgroupMap for better definitional equalities.

                                                                                                  Equations
                                                                                                  • H.equivMapOfInjective f hf = { toEquiv := Equiv.Set.image (⇑f) (↑H) hf, map_mul' := }
                                                                                                  Instances For
                                                                                                    noncomputable def AddSubgroup.equivMapOfInjective {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (f : G →+ N) (hf : Function.Injective f) :
                                                                                                    H ≃+ (AddSubgroup.map f H)

                                                                                                    An additive subgroup is isomorphic to its image under an injective function. If you have an isomorphism, use AddEquiv.addSubgroupMap for better definitional equalities.

                                                                                                    Equations
                                                                                                    • H.equivMapOfInjective f hf = { toEquiv := Equiv.Set.image (⇑f) (↑H) hf, map_add' := }
                                                                                                    Instances For
                                                                                                      @[simp]
                                                                                                      theorem Subgroup.coe_equivMapOfInjective_apply {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) (f : G →* N) (hf : Function.Injective f) (h : H) :
                                                                                                      ((H.equivMapOfInjective f hf) h) = f h
                                                                                                      @[simp]
                                                                                                      theorem AddSubgroup.coe_equivMapOfInjective_apply {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (f : G →+ N) (hf : Function.Injective f) (h : H) :
                                                                                                      ((H.equivMapOfInjective f hf) h) = f h
                                                                                                      theorem Subgroup.comap_normalizer_eq_of_surjective {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) {f : N →* G} (hf : Function.Surjective f) :
                                                                                                      Subgroup.comap f H.normalizer = (Subgroup.comap f H).normalizer

                                                                                                      The preimage of the normalizer is equal to the normalizer of the preimage of a surjective function.

                                                                                                      theorem AddSubgroup.comap_normalizer_eq_of_surjective {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) {f : N →+ G} (hf : Function.Surjective f) :
                                                                                                      AddSubgroup.comap f H.normalizer = (AddSubgroup.comap f H).normalizer

                                                                                                      The preimage of the normalizer is equal to the normalizer of the preimage of a surjective function.

                                                                                                      theorem Subgroup.comap_normalizer_eq_of_injective_of_le_range {G : Type u_1} [Group G] {N : Type u_6} [Group N] (H : Subgroup G) {f : N →* G} (hf : Function.Injective f) (h : H.normalizer f.range) :
                                                                                                      Subgroup.comap f H.normalizer = (Subgroup.comap f H).normalizer
                                                                                                      theorem AddSubgroup.comap_normalizer_eq_of_injective_of_le_range {G : Type u_1} [AddGroup G] {N : Type u_6} [AddGroup N] (H : AddSubgroup G) {f : N →+ G} (hf : Function.Injective f) (h : H.normalizer f.range) :
                                                                                                      AddSubgroup.comap f H.normalizer = (AddSubgroup.comap f H).normalizer
                                                                                                      theorem Subgroup.subgroupOf_normalizer_eq {G : Type u_1} [Group G] {H N : Subgroup G} (h : H.normalizer N) :
                                                                                                      H.normalizer.subgroupOf N = (H.subgroupOf N).normalizer
                                                                                                      theorem AddSubgroup.addSubgroupOf_normalizer_eq {G : Type u_1} [AddGroup G] {H N : AddSubgroup G} (h : H.normalizer N) :
                                                                                                      H.normalizer.addSubgroupOf N = (H.addSubgroupOf N).normalizer
                                                                                                      theorem Subgroup.map_equiv_normalizer_eq {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) (f : G ≃* N) :
                                                                                                      Subgroup.map f.toMonoidHom H.normalizer = (Subgroup.map f.toMonoidHom H).normalizer

                                                                                                      The image of the normalizer is equal to the normalizer of the image of an isomorphism.

                                                                                                      theorem AddSubgroup.map_equiv_normalizer_eq {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (f : G ≃+ N) :
                                                                                                      AddSubgroup.map f.toAddMonoidHom H.normalizer = (AddSubgroup.map f.toAddMonoidHom H).normalizer

                                                                                                      The image of the normalizer is equal to the normalizer of the image of an isomorphism.

                                                                                                      theorem Subgroup.map_normalizer_eq_of_bijective {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) {f : G →* N} (hf : Function.Bijective f) :
                                                                                                      Subgroup.map f H.normalizer = (Subgroup.map f H).normalizer

                                                                                                      The image of the normalizer is equal to the normalizer of the image of a bijective function.

                                                                                                      theorem AddSubgroup.map_normalizer_eq_of_bijective {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) {f : G →+ N} (hf : Function.Bijective f) :
                                                                                                      AddSubgroup.map f H.normalizer = (AddSubgroup.map f H).normalizer

                                                                                                      The image of the normalizer is equal to the normalizer of the image of a bijective function.

                                                                                                      def MonoidHom.liftOfRightInverseAux {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [Group G₁] [Group G₂] [Group G₃] (f : G₁ →* G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : G₁ →* G₃) (hg : f.ker g.ker) :
                                                                                                      G₂ →* G₃

                                                                                                      Auxiliary definition used to define liftOfRightInverse

                                                                                                      Equations
                                                                                                      • f.liftOfRightInverseAux f_inv hf g hg = { toFun := fun (b : G₂) => g (f_inv b), map_one' := , map_mul' := }
                                                                                                      Instances For
                                                                                                        def AddMonoidHom.liftOfRightInverseAux {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [AddGroup G₁] [AddGroup G₂] [AddGroup G₃] (f : G₁ →+ G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : G₁ →+ G₃) (hg : f.ker g.ker) :
                                                                                                        G₂ →+ G₃

                                                                                                        Auxiliary definition used to define liftOfRightInverse

                                                                                                        Equations
                                                                                                        • f.liftOfRightInverseAux f_inv hf g hg = { toFun := fun (b : G₂) => g (f_inv b), map_zero' := , map_add' := }
                                                                                                        Instances For
                                                                                                          @[simp]
                                                                                                          theorem MonoidHom.liftOfRightInverseAux_comp_apply {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [Group G₁] [Group G₂] [Group G₃] (f : G₁ →* G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : G₁ →* G₃) (hg : f.ker g.ker) (x : G₁) :
                                                                                                          (f.liftOfRightInverseAux f_inv hf g hg) (f x) = g x
                                                                                                          @[simp]
                                                                                                          theorem AddMonoidHom.liftOfRightInverseAux_comp_apply {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [AddGroup G₁] [AddGroup G₂] [AddGroup G₃] (f : G₁ →+ G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : G₁ →+ G₃) (hg : f.ker g.ker) (x : G₁) :
                                                                                                          (f.liftOfRightInverseAux f_inv hf g hg) (f x) = g x
                                                                                                          def MonoidHom.liftOfRightInverse {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [Group G₁] [Group G₂] [Group G₃] (f : G₁ →* G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) :
                                                                                                          { g : G₁ →* G₃ // f.ker g.ker } (G₂ →* G₃)

                                                                                                          liftOfRightInverse f hf g hg is the unique group homomorphism φ

                                                                                                          See MonoidHom.eq_liftOfRightInverse for the uniqueness lemma.

                                                                                                             G₁.
                                                                                                             |  \
                                                                                                           f |   \ g
                                                                                                             |    \
                                                                                                             v     \⌟
                                                                                                             G₂----> G₃
                                                                                                                ∃!φ
                                                                                                          
                                                                                                          Equations
                                                                                                          • One or more equations did not get rendered due to their size.
                                                                                                          Instances For
                                                                                                            def AddMonoidHom.liftOfRightInverse {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [AddGroup G₁] [AddGroup G₂] [AddGroup G₃] (f : G₁ →+ G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) :
                                                                                                            { g : G₁ →+ G₃ // f.ker g.ker } (G₂ →+ G₃)

                                                                                                            liftOfRightInverse f f_inv hf g hg is the unique additive group homomorphism φ

                                                                                                               G₁.
                                                                                                               |  \
                                                                                                             f |   \ g
                                                                                                               |    \
                                                                                                               v     \⌟
                                                                                                               G₂----> G₃
                                                                                                                  ∃!φ
                                                                                                            
                                                                                                            Equations
                                                                                                            • One or more equations did not get rendered due to their size.
                                                                                                            Instances For
                                                                                                              @[reducible, inline]
                                                                                                              noncomputable abbrev MonoidHom.liftOfSurjective {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [Group G₁] [Group G₂] [Group G₃] (f : G₁ →* G₂) (hf : Function.Surjective f) :
                                                                                                              { g : G₁ →* G₃ // f.ker g.ker } (G₂ →* G₃)

                                                                                                              A non-computable version of MonoidHom.liftOfRightInverse for when no computable right inverse is available, that uses Function.surjInv.

                                                                                                              Equations
                                                                                                              Instances For
                                                                                                                @[reducible, inline]
                                                                                                                noncomputable abbrev AddMonoidHom.liftOfSurjective {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [AddGroup G₁] [AddGroup G₂] [AddGroup G₃] (f : G₁ →+ G₂) (hf : Function.Surjective f) :
                                                                                                                { g : G₁ →+ G₃ // f.ker g.ker } (G₂ →+ G₃)

                                                                                                                A non-computable version of AddMonoidHom.liftOfRightInverse for when no computable right inverse is available.

                                                                                                                Equations
                                                                                                                Instances For
                                                                                                                  @[simp]
                                                                                                                  theorem MonoidHom.liftOfRightInverse_comp_apply {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [Group G₁] [Group G₂] [Group G₃] (f : G₁ →* G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : { g : G₁ →* G₃ // f.ker g.ker }) (x : G₁) :
                                                                                                                  ((f.liftOfRightInverse f_inv hf) g) (f x) = g x
                                                                                                                  @[simp]
                                                                                                                  theorem AddMonoidHom.liftOfRightInverse_comp_apply {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [AddGroup G₁] [AddGroup G₂] [AddGroup G₃] (f : G₁ →+ G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : { g : G₁ →+ G₃ // f.ker g.ker }) (x : G₁) :
                                                                                                                  ((f.liftOfRightInverse f_inv hf) g) (f x) = g x
                                                                                                                  @[simp]
                                                                                                                  theorem MonoidHom.liftOfRightInverse_comp {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [Group G₁] [Group G₂] [Group G₃] (f : G₁ →* G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : { g : G₁ →* G₃ // f.ker g.ker }) :
                                                                                                                  ((f.liftOfRightInverse f_inv hf) g).comp f = g
                                                                                                                  @[simp]
                                                                                                                  theorem AddMonoidHom.liftOfRightInverse_comp {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [AddGroup G₁] [AddGroup G₂] [AddGroup G₃] (f : G₁ →+ G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : { g : G₁ →+ G₃ // f.ker g.ker }) :
                                                                                                                  ((f.liftOfRightInverse f_inv hf) g).comp f = g
                                                                                                                  theorem MonoidHom.eq_liftOfRightInverse {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [Group G₁] [Group G₂] [Group G₃] (f : G₁ →* G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : G₁ →* G₃) (hg : f.ker g.ker) (h : G₂ →* G₃) (hh : h.comp f = g) :
                                                                                                                  h = (f.liftOfRightInverse f_inv hf) g, hg
                                                                                                                  theorem AddMonoidHom.eq_liftOfRightInverse {G₁ : Type u_5} {G₂ : Type u_6} {G₃ : Type u_7} [AddGroup G₁] [AddGroup G₂] [AddGroup G₃] (f : G₁ →+ G₂) (f_inv : G₂G₁) (hf : Function.RightInverse f_inv f) (g : G₁ →+ G₃) (hg : f.ker g.ker) (h : G₂ →+ G₃) (hh : h.comp f = g) :
                                                                                                                  h = (f.liftOfRightInverse f_inv hf) g, hg
                                                                                                                  theorem Subgroup.Normal.comap {G : Type u_1} [Group G] {N : Type u_5} [Group N] {H : Subgroup N} (hH : H.Normal) (f : G →* N) :
                                                                                                                  (Subgroup.comap f H).Normal
                                                                                                                  theorem AddSubgroup.Normal.comap {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup N} (hH : H.Normal) (f : G →+ N) :
                                                                                                                  (AddSubgroup.comap f H).Normal
                                                                                                                  @[instance 100]
                                                                                                                  instance Subgroup.normal_comap {G : Type u_1} [Group G] {N : Type u_5} [Group N] {H : Subgroup N} [nH : H.Normal] (f : G →* N) :
                                                                                                                  (Subgroup.comap f H).Normal
                                                                                                                  Equations
                                                                                                                  • =
                                                                                                                  @[instance 100]
                                                                                                                  instance AddSubgroup.normal_comap {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup N} [nH : H.Normal] (f : G →+ N) :
                                                                                                                  (AddSubgroup.comap f H).Normal
                                                                                                                  Equations
                                                                                                                  • =
                                                                                                                  theorem Subgroup.Normal.subgroupOf {G : Type u_1} [Group G] {H : Subgroup G} (hH : H.Normal) (K : Subgroup G) :
                                                                                                                  (H.subgroupOf K).Normal
                                                                                                                  theorem AddSubgroup.Normal.addSubgroupOf {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (hH : H.Normal) (K : AddSubgroup G) :
                                                                                                                  (H.addSubgroupOf K).Normal
                                                                                                                  @[instance 100]
                                                                                                                  instance Subgroup.normal_subgroupOf {G : Type u_1} [Group G] {H N : Subgroup G} [N.Normal] :
                                                                                                                  (N.subgroupOf H).Normal
                                                                                                                  Equations
                                                                                                                  • =
                                                                                                                  @[instance 100]
                                                                                                                  instance AddSubgroup.normal_addSubgroupOf {G : Type u_1} [AddGroup G] {H N : AddSubgroup G} [N.Normal] :
                                                                                                                  (N.addSubgroupOf H).Normal
                                                                                                                  Equations
                                                                                                                  • =
                                                                                                                  theorem Subgroup.map_normalClosure {G : Type u_1} [Group G] {N : Type u_5} [Group N] (s : Set G) (f : G →* N) (hf : Function.Surjective f) :
                                                                                                                  theorem Subgroup.comap_normalClosure {G : Type u_1} [Group G] {N : Type u_5} [Group N] (s : Set N) (f : G ≃* N) :
                                                                                                                  theorem Subgroup.Normal.of_map_injective {G : Type u_6} {H : Type u_7} [Group G] [Group H] {φ : G →* H} (hφ : Function.Injective φ) {L : Subgroup G} (n : (Subgroup.map φ L).Normal) :
                                                                                                                  L.Normal
                                                                                                                  theorem Subgroup.Normal.of_map_subtype {G : Type u_1} [Group G] {K : Subgroup G} {L : Subgroup K} (n : (Subgroup.map K.subtype L).Normal) :
                                                                                                                  L.Normal
                                                                                                                  def MonoidHom.subgroupComap {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') (H' : Subgroup G') :
                                                                                                                  (Subgroup.comap f H') →* H'

                                                                                                                  The MonoidHom from the preimage of a subgroup to itself.

                                                                                                                  Equations
                                                                                                                  • f.subgroupComap H' = f.submonoidComap H'.toSubmonoid
                                                                                                                  Instances For
                                                                                                                    def AddMonoidHom.addSubgroupComap {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H' : AddSubgroup G') :
                                                                                                                    (AddSubgroup.comap f H') →+ H'

                                                                                                                    the AddMonoidHom from the preimage of an additive subgroup to itself.

                                                                                                                    Equations
                                                                                                                    • f.addSubgroupComap H' = f.addSubmonoidComap H'.toAddSubmonoid
                                                                                                                    Instances For
                                                                                                                      @[simp]
                                                                                                                      theorem AddMonoidHom.addSubgroupComap_apply_coe {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H' : AddSubgroup G') (x : (AddSubmonoid.comap f H'.toAddSubmonoid)) :
                                                                                                                      ((f.addSubgroupComap H') x) = f x
                                                                                                                      @[simp]
                                                                                                                      theorem MonoidHom.subgroupComap_apply_coe {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') (H' : Subgroup G') (x : (Submonoid.comap f H'.toSubmonoid)) :
                                                                                                                      ((f.subgroupComap H') x) = f x
                                                                                                                      def MonoidHom.subgroupMap {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') (H : Subgroup G) :
                                                                                                                      H →* (Subgroup.map f H)

                                                                                                                      The MonoidHom from a subgroup to its image.

                                                                                                                      Equations
                                                                                                                      • f.subgroupMap H = f.submonoidMap H.toSubmonoid
                                                                                                                      Instances For
                                                                                                                        def AddMonoidHom.addSubgroupMap {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H : AddSubgroup G) :
                                                                                                                        H →+ (AddSubgroup.map f H)

                                                                                                                        the AddMonoidHom from an additive subgroup to its image

                                                                                                                        Equations
                                                                                                                        • f.addSubgroupMap H = f.addSubmonoidMap H.toAddSubmonoid
                                                                                                                        Instances For
                                                                                                                          @[simp]
                                                                                                                          theorem MonoidHom.subgroupMap_apply_coe {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') (H : Subgroup G) (x : H.toSubmonoid) :
                                                                                                                          ((f.subgroupMap H) x) = f x
                                                                                                                          @[simp]
                                                                                                                          theorem AddMonoidHom.addSubgroupMap_apply_coe {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H : AddSubgroup G) (x : H.toAddSubmonoid) :
                                                                                                                          ((f.addSubgroupMap H) x) = f x
                                                                                                                          theorem MonoidHom.subgroupMap_surjective {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') (H : Subgroup G) :
                                                                                                                          Function.Surjective (f.subgroupMap H)
                                                                                                                          theorem AddMonoidHom.addSubgroupMap_surjective {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H : AddSubgroup G) :
                                                                                                                          Function.Surjective (f.addSubgroupMap H)
                                                                                                                          def MulEquiv.subgroupCongr {G : Type u_1} [Group G] {H K : Subgroup G} (h : H = K) :
                                                                                                                          H ≃* K

                                                                                                                          Makes the identity isomorphism from a proof two subgroups of a multiplicative group are equal.

                                                                                                                          Equations
                                                                                                                          Instances For
                                                                                                                            def AddEquiv.addSubgroupCongr {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H = K) :
                                                                                                                            H ≃+ K

                                                                                                                            Makes the identity additive isomorphism from a proof two subgroups of an additive group are equal.

                                                                                                                            Equations
                                                                                                                            Instances For
                                                                                                                              @[simp]
                                                                                                                              theorem MulEquiv.subgroupCongr_apply {G : Type u_1} [Group G] {H K : Subgroup G} (h : H = K) (x : H) :
                                                                                                                              ((MulEquiv.subgroupCongr h) x) = x
                                                                                                                              @[simp]
                                                                                                                              theorem AddEquiv.addSubgroupCongr_apply {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H = K) (x : H) :
                                                                                                                              @[simp]
                                                                                                                              theorem MulEquiv.subgroupCongr_symm_apply {G : Type u_1} [Group G] {H K : Subgroup G} (h : H = K) (x : K) :
                                                                                                                              ((MulEquiv.subgroupCongr h).symm x) = x
                                                                                                                              @[simp]
                                                                                                                              theorem AddEquiv.addSubgroupCongr_symm_apply {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H = K) (x : K) :
                                                                                                                              ((AddEquiv.addSubgroupCongr h).symm x) = x
                                                                                                                              def MulEquiv.subgroupMap {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (e : G ≃* G') (H : Subgroup G) :
                                                                                                                              H ≃* (Subgroup.map (↑e) H)

                                                                                                                              A subgroup is isomorphic to its image under an isomorphism. If you only have an injective map, use Subgroup.equiv_map_of_injective.

                                                                                                                              Equations
                                                                                                                              • e.subgroupMap H = e.submonoidMap H.toSubmonoid
                                                                                                                              Instances For
                                                                                                                                def AddEquiv.addSubgroupMap {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (e : G ≃+ G') (H : AddSubgroup G) :
                                                                                                                                H ≃+ (AddSubgroup.map (↑e) H)

                                                                                                                                An additive subgroup is isomorphic to its image under an isomorphism. If you only have an injective map, use AddSubgroup.equiv_map_of_injective.

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                                                                                                                                • e.addSubgroupMap H = e.addSubmonoidMap H.toAddSubmonoid
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                                                                                                                                  @[simp]
                                                                                                                                  theorem MulEquiv.coe_subgroupMap_apply {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (e : G ≃* G') (H : Subgroup G) (g : H) :
                                                                                                                                  ((e.subgroupMap H) g) = e g
                                                                                                                                  @[simp]
                                                                                                                                  theorem AddEquiv.coe_addSubgroupMap_apply {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (e : G ≃+ G') (H : AddSubgroup G) (g : H) :
                                                                                                                                  ((e.addSubgroupMap H) g) = e g
                                                                                                                                  @[simp]
                                                                                                                                  theorem MulEquiv.subgroupMap_symm_apply {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (e : G ≃* G') (H : Subgroup G) (g : (Subgroup.map (↑e) H)) :
                                                                                                                                  (e.subgroupMap H).symm g = e.symm g,
                                                                                                                                  @[simp]
                                                                                                                                  theorem AddEquiv.addSubgroupMap_symm_apply {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (e : G ≃+ G') (H : AddSubgroup G) (g : (AddSubgroup.map (↑e) H)) :
                                                                                                                                  (e.addSubgroupMap H).symm g = e.symm g,
                                                                                                                                  @[simp]
                                                                                                                                  theorem Subgroup.equivMapOfInjective_coe_mulEquiv {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) (e : G ≃* G') :
                                                                                                                                  H.equivMapOfInjective e = e.subgroupMap H
                                                                                                                                  @[simp]
                                                                                                                                  theorem AddSubgroup.equivMapOfInjective_coe_addEquiv {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (e : G ≃+ G') :
                                                                                                                                  H.equivMapOfInjective e = e.addSubgroupMap H
                                                                                                                                  theorem Subgroup.mem_sup {C : Type u_6} [CommGroup C] {s t : Subgroup C} {x : C} :
                                                                                                                                  x s t ys, zt, y * z = x
                                                                                                                                  theorem AddSubgroup.mem_sup {C : Type u_6} [AddCommGroup C] {s t : AddSubgroup C} {x : C} :
                                                                                                                                  x s t ys, zt, y + z = x
                                                                                                                                  theorem Subgroup.mem_sup' {C : Type u_6} [CommGroup C] {s t : Subgroup C} {x : C} :
                                                                                                                                  x s t ∃ (y : s) (z : t), y * z = x
                                                                                                                                  theorem AddSubgroup.mem_sup' {C : Type u_6} [AddCommGroup C] {s t : AddSubgroup C} {x : C} :
                                                                                                                                  x s t ∃ (y : s) (z : t), y + z = x
                                                                                                                                  theorem Subgroup.mem_closure_pair {C : Type u_6} [CommGroup C] {x y z : C} :
                                                                                                                                  z Subgroup.closure {x, y} ∃ (m : ) (n : ), x ^ m * y ^ n = z
                                                                                                                                  theorem AddSubgroup.mem_closure_pair {C : Type u_6} [AddCommGroup C] {x y z : C} :
                                                                                                                                  z AddSubgroup.closure {x, y} ∃ (m : ) (n : ), m x + n y = z
                                                                                                                                  theorem Subgroup.normal_subgroupOf_iff {G : Type u_1} [Group G] {H K : Subgroup G} (hHK : H K) :
                                                                                                                                  (H.subgroupOf K).Normal ∀ (h k : G), h Hk Kk * h * k⁻¹ H
                                                                                                                                  theorem AddSubgroup.normal_addSubgroupOf_iff {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (hHK : H K) :
                                                                                                                                  (H.addSubgroupOf K).Normal ∀ (h k : G), h Hk Kk + h + -k H
                                                                                                                                  instance Subgroup.prod_subgroupOf_prod_normal {G : Type u_1} [Group G] {N : Type u_5} [Group N] {H₁ K₁ : Subgroup G} {H₂ K₂ : Subgroup N} [h₁ : (H₁.subgroupOf K₁).Normal] [h₂ : (H₂.subgroupOf K₂).Normal] :
                                                                                                                                  ((H₁.prod H₂).subgroupOf (K₁.prod K₂)).Normal
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                                                                                                                                  instance AddSubgroup.sum_addSubgroupOf_sum_normal {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H₁ K₁ : AddSubgroup G} {H₂ K₂ : AddSubgroup N} [h₁ : (H₁.addSubgroupOf K₁).Normal] [h₂ : (H₂.addSubgroupOf K₂).Normal] :
                                                                                                                                  ((H₁.prod H₂).addSubgroupOf (K₁.prod K₂)).Normal
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                                                                                                                                  instance Subgroup.prod_normal {G : Type u_1} [Group G] {N : Type u_5} [Group N] (H : Subgroup G) (K : Subgroup N) [hH : H.Normal] [hK : K.Normal] :
                                                                                                                                  (H.prod K).Normal
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                                                                                                                                  instance AddSubgroup.sum_normal {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (K : AddSubgroup N) [hH : H.Normal] [hK : K.Normal] :
                                                                                                                                  (H.prod K).Normal
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                                                                                                                                  theorem Subgroup.inf_subgroupOf_inf_normal_of_right {G : Type u_1} [Group G] (A B' B : Subgroup G) (hB : B' B) [hN : (B'.subgroupOf B).Normal] :
                                                                                                                                  ((A B').subgroupOf (A B)).Normal
                                                                                                                                  theorem AddSubgroup.inf_addSubgroupOf_inf_normal_of_right {G : Type u_1} [AddGroup G] (A B' B : AddSubgroup G) (hB : B' B) [hN : (B'.addSubgroupOf B).Normal] :
                                                                                                                                  ((A B').addSubgroupOf (A B)).Normal
                                                                                                                                  theorem Subgroup.inf_subgroupOf_inf_normal_of_left {G : Type u_1} [Group G] {A' A : Subgroup G} (B : Subgroup G) (hA : A' A) [hN : (A'.subgroupOf A).Normal] :
                                                                                                                                  ((A' B).subgroupOf (A B)).Normal
                                                                                                                                  theorem AddSubgroup.inf_addSubgroupOf_inf_normal_of_left {G : Type u_1} [AddGroup G] {A' A : AddSubgroup G} (B : AddSubgroup G) (hA : A' A) [hN : (A'.addSubgroupOf A).Normal] :
                                                                                                                                  ((A' B).addSubgroupOf (A B)).Normal
                                                                                                                                  instance Subgroup.normal_inf_normal {G : Type u_1} [Group G] (H K : Subgroup G) [hH : H.Normal] [hK : K.Normal] :
                                                                                                                                  (H K).Normal
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                                                                                                                                  instance AddSubgroup.normal_inf_normal {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) [hH : H.Normal] [hK : K.Normal] :
                                                                                                                                  (H K).Normal
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                                                                                                                                  theorem Subgroup.subgroupOf_sup {G : Type u_1} [Group G] (A A' B : Subgroup G) (hA : A B) (hA' : A' B) :
                                                                                                                                  (A A').subgroupOf B = A.subgroupOf B A'.subgroupOf B
                                                                                                                                  theorem AddSubgroup.addSubgroupOf_sup {G : Type u_1} [AddGroup G] (A A' B : AddSubgroup G) (hA : A B) (hA' : A' B) :
                                                                                                                                  (A A').addSubgroupOf B = A.addSubgroupOf B A'.addSubgroupOf B
                                                                                                                                  theorem Subgroup.SubgroupNormal.mem_comm {G : Type u_1} [Group G] {H K : Subgroup G} (hK : H K) [hN : (H.subgroupOf K).Normal] {a b : G} (hb : b K) (h : a * b H) :
                                                                                                                                  b * a H
                                                                                                                                  theorem AddSubgroup.SubgroupNormal.mem_comm {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (hK : H K) [hN : (H.addSubgroupOf K).Normal] {a b : G} (hb : b K) (h : a + b H) :
                                                                                                                                  b + a H
                                                                                                                                  theorem Subgroup.commute_of_normal_of_disjoint {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) (hH₁ : H₁.Normal) (hH₂ : H₂.Normal) (hdis : Disjoint H₁ H₂) (x y : G) (hx : x H₁) (hy : y H₂) :

                                                                                                                                  Elements of disjoint, normal subgroups commute.

                                                                                                                                  theorem AddSubgroup.addCommute_of_normal_of_disjoint {G : Type u_1} [AddGroup G] (H₁ H₂ : AddSubgroup G) (hH₁ : H₁.Normal) (hH₂ : H₂.Normal) (hdis : Disjoint H₁ H₂) (x y : G) (hx : x H₁) (hy : y H₂) :

                                                                                                                                  Elements of disjoint, normal subgroups commute.

                                                                                                                                  theorem Subgroup.disjoint_def {G : Type u_1} [Group G] {H₁ H₂ : Subgroup G} :
                                                                                                                                  Disjoint H₁ H₂ ∀ {x : G}, x H₁x H₂x = 1
                                                                                                                                  theorem AddSubgroup.disjoint_def {G : Type u_1} [AddGroup G] {H₁ H₂ : AddSubgroup G} :
                                                                                                                                  Disjoint H₁ H₂ ∀ {x : G}, x H₁x H₂x = 0
                                                                                                                                  theorem Subgroup.disjoint_def' {G : Type u_1} [Group G] {H₁ H₂ : Subgroup G} :
                                                                                                                                  Disjoint H₁ H₂ ∀ {x y : G}, x H₁y H₂x = yx = 1
                                                                                                                                  theorem AddSubgroup.disjoint_def' {G : Type u_1} [AddGroup G] {H₁ H₂ : AddSubgroup G} :
                                                                                                                                  Disjoint H₁ H₂ ∀ {x y : G}, x H₁y H₂x = yx = 0
                                                                                                                                  theorem Subgroup.disjoint_iff_mul_eq_one {G : Type u_1} [Group G] {H₁ H₂ : Subgroup G} :
                                                                                                                                  Disjoint H₁ H₂ ∀ {x y : G}, x H₁y H₂x * y = 1x = 1 y = 1
                                                                                                                                  theorem AddSubgroup.disjoint_iff_add_eq_zero {G : Type u_1} [AddGroup G] {H₁ H₂ : AddSubgroup G} :
                                                                                                                                  Disjoint H₁ H₂ ∀ {x y : G}, x H₁y H₂x + y = 0x = 0 y = 0
                                                                                                                                  theorem Subgroup.mul_injective_of_disjoint {G : Type u_1} [Group G] {H₁ H₂ : Subgroup G} (h : Disjoint H₁ H₂) :
                                                                                                                                  Function.Injective fun (g : H₁ × H₂) => g.1 * g.2
                                                                                                                                  theorem AddSubgroup.add_injective_of_disjoint {G : Type u_1} [AddGroup G] {H₁ H₂ : AddSubgroup G} (h : Disjoint H₁ H₂) :
                                                                                                                                  Function.Injective fun (g : H₁ × H₂) => g.1 + g.2
                                                                                                                                  theorem IsConj.normalClosure_eq_top_of {G : Type u_1} [Group G] {N : Subgroup G} [hn : N.Normal] {g g' : G} {hg : g N} {hg' : g' N} (hc : IsConj g g') (ht : Subgroup.normalClosure {g, hg} = ) :
                                                                                                                                  Subgroup.normalClosure {g', hg'} =

                                                                                                                                  The conjugacy classes that are not trivial.

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                                                                                                                                    theorem ConjClasses.mem_noncenter {G : Type u_6} [Monoid G] (g : ConjClasses G) :
                                                                                                                                    g ConjClasses.noncenter G g.carrier.Nontrivial