Documentation

Mathlib.Algebra.Group.Subgroup.MulOpposite

Mul-opposite subgroups #

Tags #

subgroup, subgroups

source
def Subgroup.op {G : Type u_2} [Group G] (H : Subgroup G) :
Subgroup Gᵐᵒᵖ

Pull a subgroup back to an opposite subgroup along MulOpposite.unop

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      def AddSubgroup.op {G : Type u_2} [AddGroup G] (H : AddSubgroup G) :
      AddSubgroup Gᵃᵒᵖ

      Pull an additive subgroup back to an opposite additive subgroup along AddOpposite.unop

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          @[simp]
          theorem AddSubgroup.coe_op {G : Type u_2} [AddGroup G] (H : AddSubgroup G) :
          ↑H.op = AddOpposite.unop ⁻¹' ↑H
          source
          @[simp]
          theorem Subgroup.coe_op {G : Type u_2} [Group G] (H : Subgroup G) :
          ↑H.op = MulOpposite.unop ⁻¹' ↑H
          source
          @[simp]
          theorem Subgroup.mem_op {G : Type u_2} [Group G] {x : Gᵐᵒᵖ} {S : Subgroup G} :
          x ∈ S.op ↔ MulOpposite.unop x ∈ S
          source
          @[simp]
          theorem AddSubgroup.mem_op {G : Type u_2} [AddGroup G] {x : Gᵃᵒᵖ} {S : AddSubgroup G} :
          x ∈ S.op ↔ AddOpposite.unop x ∈ S
          source
          @[simp]
          theorem Subgroup.op_toSubmonoid {G : Type u_2} [Group G] (H : Subgroup G) :
          H.op.toSubmonoid = H.op
          source
          @[simp]
          theorem AddSubgroup.op_toAddSubmonoid {G : Type u_2} [AddGroup G] (H : AddSubgroup G) :
          H.op.toAddSubmonoid = H.op
          source
          def Subgroup.unop {G : Type u_2} [Group G] (H : Subgroup Gᵐᵒᵖ) :
          Subgroup G

          Pull an opposite subgroup back to a subgroup along MulOpposite.op

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              def AddSubgroup.unop {G : Type u_2} [AddGroup G] (H : AddSubgroup Gᵃᵒᵖ) :
              AddSubgroup G

              Pull an opposite additive subgroup back to an additive subgroup along AddOpposite.op

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                  @[simp]
                  theorem Subgroup.coe_unop {G : Type u_2} [Group G] (H : Subgroup Gᵐᵒᵖ) :
                  ↑H.unop = MulOpposite.op ⁻¹' ↑H
                  source
                  @[simp]
                  theorem AddSubgroup.coe_unop {G : Type u_2} [AddGroup G] (H : AddSubgroup Gᵃᵒᵖ) :
                  ↑H.unop = AddOpposite.op ⁻¹' ↑H
                  source
                  @[simp]
                  theorem Subgroup.mem_unop {G : Type u_2} [Group G] {x : G} {S : Subgroup Gᵐᵒᵖ} :
                  x ∈ S.unop ↔ MulOpposite.op x ∈ S
                  source
                  @[simp]
                  theorem AddSubgroup.mem_unop {G : Type u_2} [AddGroup G] {x : G} {S : AddSubgroup Gᵃᵒᵖ} :
                  x ∈ S.unop ↔ AddOpposite.op x ∈ S
                  source
                  @[simp]
                  theorem Subgroup.unop_toSubmonoid {G : Type u_2} [Group G] (H : Subgroup Gᵐᵒᵖ) :
                  H.unop.toSubmonoid = H.unop
                  source
                  @[simp]
                  theorem AddSubgroup.unop_toAddSubmonoid {G : Type u_2} [AddGroup G] (H : AddSubgroup Gᵃᵒᵖ) :
                  H.unop.toAddSubmonoid = H.unop
                  source
                  @[simp]
                  theorem Subgroup.unop_op {G : Type u_2} [Group G] (S : Subgroup G) :
                  S.op.unop = S
                  source
                  @[simp]
                  theorem AddSubgroup.unop_op {G : Type u_2} [AddGroup G] (S : AddSubgroup G) :
                  S.op.unop = S
                  source
                  @[simp]
                  theorem Subgroup.op_unop {G : Type u_2} [Group G] (S : Subgroup Gᵐᵒᵖ) :
                  S.unop.op = S
                  source
                  @[simp]
                  theorem AddSubgroup.op_unop {G : Type u_2} [AddGroup G] (S : AddSubgroup Gᵃᵒᵖ) :
                  S.unop.op = S

                  Lattice results #

                  source
                  theorem Subgroup.op_le_iff {G : Type u_2} [Group G] {S₁ : Subgroup G} {S₂ : Subgroup Gᵐᵒᵖ} :
                  S₁.op ≤ S₂ ↔ S₁ ≤ S₂.unop
                  source
                  theorem AddSubgroup.op_le_iff {G : Type u_2} [AddGroup G] {S₁ : AddSubgroup G} {S₂ : AddSubgroup Gᵃᵒᵖ} :
                  S₁.op ≤ S₂ ↔ S₁ ≤ S₂.unop
                  source
                  theorem Subgroup.le_op_iff {G : Type u_2} [Group G] {S₁ : Subgroup Gᵐᵒᵖ} {S₂ : Subgroup G} :
                  S₁ ≤ S₂.op ↔ S₁.unop ≤ S₂
                  source
                  theorem AddSubgroup.le_op_iff {G : Type u_2} [AddGroup G] {S₁ : AddSubgroup Gᵃᵒᵖ} {S₂ : AddSubgroup G} :
                  S₁ ≤ S₂.op ↔ S₁.unop ≤ S₂
                  source
                  @[simp]
                  theorem Subgroup.op_le_op_iff {G : Type u_2} [Group G] {S₁ S₂ : Subgroup G} :
                  S₁.op ≤ S₂.op ↔ S₁ ≤ S₂
                  source
                  @[simp]
                  theorem AddSubgroup.op_le_op_iff {G : Type u_2} [AddGroup G] {S₁ S₂ : AddSubgroup G} :
                  S₁.op ≤ S₂.op ↔ S₁ ≤ S₂
                  source
                  @[simp]
                  theorem Subgroup.unop_le_unop_iff {G : Type u_2} [Group G] {S₁ S₂ : Subgroup Gᵐᵒᵖ} :
                  S₁.unop ≤ S₂.unop ↔ S₁ ≤ S₂
                  source
                  @[simp]
                  theorem AddSubgroup.unop_le_unop_iff {G : Type u_2} [AddGroup G] {S₁ S₂ : AddSubgroup Gᵃᵒᵖ} :
                  S₁.unop ≤ S₂.unop ↔ S₁ ≤ S₂
                  source
                  def Subgroup.opEquiv {G : Type u_2} [Group G] :
                  Subgroup G ≃o Subgroup Gᵐᵒᵖ

                  A subgroup H of G determines a subgroup H.op of the opposite group Gᵐᵒᵖ.

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                      def AddSubgroup.opEquiv {G : Type u_2} [AddGroup G] :
                      AddSubgroup G ≃o AddSubgroup Gᵃᵒᵖ

                      An additive subgroup H of G determines an additive subgroup H.op of the opposite additive group Gᵃᵒᵖ.

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                          @[simp]
                          theorem AddSubgroup.opEquiv_apply {G : Type u_2} [AddGroup G] (H : AddSubgroup G) :
                          opEquiv H = H.op
                          source
                          @[simp]
                          theorem AddSubgroup.opEquiv_symm_apply {G : Type u_2} [AddGroup G] (H : AddSubgroup Gᵃᵒᵖ) :
                          (RelIso.symm opEquiv) H = H.unop
                          source
                          @[simp]
                          theorem Subgroup.opEquiv_apply {G : Type u_2} [Group G] (H : Subgroup G) :
                          opEquiv H = H.op
                          source
                          @[simp]
                          theorem Subgroup.opEquiv_symm_apply {G : Type u_2} [Group G] (H : Subgroup Gᵐᵒᵖ) :
                          (RelIso.symm opEquiv) H = H.unop
                          source
                          theorem Subgroup.op_injective {G : Type u_2} [Group G] :
                          Function.Injective Subgroup.op
                          source
                          theorem AddSubgroup.op_injective {G : Type u_2} [AddGroup G] :
                          Function.Injective AddSubgroup.op
                          source
                          theorem Subgroup.unop_injective {G : Type u_2} [Group G] :
                          Function.Injective Subgroup.unop
                          source
                          theorem AddSubgroup.unop_injective {G : Type u_2} [AddGroup G] :
                          Function.Injective AddSubgroup.unop
                          source
                          @[simp]
                          theorem Subgroup.op_inj {G : Type u_2} [Group G] {S T : Subgroup G} :
                          S.op = T.op ↔ S = T
                          source
                          @[simp]
                          theorem AddSubgroup.op_inj {G : Type u_2} [AddGroup G] {S T : AddSubgroup G} :
                          S.op = T.op ↔ S = T
                          source
                          @[simp]
                          theorem Subgroup.unop_inj {G : Type u_2} [Group G] {S T : Subgroup Gᵐᵒᵖ} :
                          S.unop = T.unop ↔ S = T
                          source
                          @[simp]
                          theorem AddSubgroup.unop_inj {G : Type u_2} [AddGroup G] {S T : AddSubgroup Gᵃᵒᵖ} :
                          S.unop = T.unop ↔ S = T
                          source
                          def Subgroup.equivOp {G : Type u_2} [Group G] (H : Subgroup G) :
                          ↥H ≃ ↥H.op

                          Bijection between a subgroup H and its opposite.

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                              def AddSubgroup.equivOp {G : Type u_2} [AddGroup G] (H : AddSubgroup G) :
                              ↥H ≃ ↥H.op

                              Bijection between an additive subgroup H and its opposite.

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                                  @[simp]
                                  theorem Subgroup.equivOp_symm_apply_coe {G : Type u_2} [Group G] (H : Subgroup G) (b : ↥H.op) :
                                  ↑(H.equivOp.symm b) = MulOpposite.unop ↑b
                                  source
                                  @[simp]
                                  theorem Subgroup.equivOp_apply_coe {G : Type u_2} [Group G] (H : Subgroup G) (a : ↥H) :
                                  ↑(H.equivOp a) = MulOpposite.op ↑a
                                  source
                                  @[simp]
                                  theorem AddSubgroup.equivOp_apply_coe {G : Type u_2} [AddGroup G] (H : AddSubgroup G) (a : ↥H) :
                                  ↑(H.equivOp a) = AddOpposite.op ↑a
                                  source
                                  @[simp]
                                  theorem AddSubgroup.equivOp_symm_apply_coe {G : Type u_2} [AddGroup G] (H : AddSubgroup G) (b : ↥H.op) :
                                  ↑(H.equivOp.symm b) = AddOpposite.unop ↑b
                                  source
                                  theorem Subgroup.op_normalizer {G : Type u_2} [Group G] (H : Subgroup G) :
                                  H.normalizer.op = H.op.normalizer
                                  source
                                  theorem AddSubgroup.op_normalizer {G : Type u_2} [AddGroup G] (H : AddSubgroup G) :
                                  H.normalizer.op = H.op.normalizer
                                  source
                                  theorem Subgroup.unop_normalizer {G : Type u_2} [Group G] (H : Subgroup Gᵐᵒᵖ) :
                                  H.normalizer.unop = H.unop.normalizer
                                  source
                                  theorem AddSubgroup.unop_normalizer {G : Type u_2} [AddGroup G] (H : AddSubgroup Gᵃᵒᵖ) :
                                  H.normalizer.unop = H.unop.normalizer