Documentation

Mathlib.GroupTheory.SemidirectProduct

Semidirect product #

This file defines semidirect products of groups, and the canonical maps in and out of the semidirect product. The semidirect product of N and G given a hom φ from G to the automorphism group of N is the product of sets with the group ⟨n₁, g₁⟩ * ⟨n₂, g₂⟩ = ⟨n₁ * φ g₁ n₂, g₁ * g₂⟩

Key definitions #

There are two homs into the semidirect product inl : N →* N ⋊[φ] G and inr : G →* N ⋊[φ] G, and lift can be used to define maps N ⋊[φ] G →* H out of the semidirect product given maps f₁ : N →* H and f₂ : G →* H that satisfy the condition ∀ n g, f₁ (φ g n) = f₂ g * f₁ n * f₂ g⁻¹

Notation #

This file introduces the global notation N ⋊[φ] G for SemidirectProduct N G φ

Tags #

group, semidirect product

source
theorem SemidirectProduct.ext {N : Type u_1} {G : Type u_2} :
∀ {inst : Group N} {inst_1 : Group G} {φ : G →* MulAut N} (x y : N ⋊[φ] G), x.left = y.left → x.right = y.right → x = y
source
theorem SemidirectProduct.ext_iff {N : Type u_1} {G : Type u_2} :
∀ {inst : Group N} {inst_1 : Group G} {φ : G →* MulAut N} (x y : N ⋊[φ] G), x = y ↔ x.left = y.left ∧ x.right = y.right
source
structure SemidirectProduct (N : Type u_1) (G : Type u_2) [Group N] [Group G] (φ : G →* MulAut N) :
Type (max u_1 u_2)
  • left : N

    The element of N

  • right : G

    The element of G

The semidirect product of groups N and G, given a map φ from G to the automorphism group of N. It the product of sets with the group operation ⟨n₁, g₁⟩ * ⟨n₂, g₂⟩ = ⟨n₁ * φ g₁ n₂, g₁ * g₂⟩

Instances For
    source
    instance instDecidableEqSemidirectProduct :
    {N : Type u_4} → {G : Type u_5} → {inst : Group N} → {inst_1 : Group G} → {φ : G →* MulAut N} → [inst_2 : DecidableEq N] → [inst_3 : DecidableEq G] → DecidableEq (N ⋊[φ] G)
    source
    def «term_⋊[_]_» :
    Lean.TrailingParserDescr

    The semidirect product of groups N and G, given a map φ from G to the automorphism group of N. It the product of sets with the group operation ⟨n₁, g₁⟩ * ⟨n₂, g₂⟩ = ⟨n₁ * φ g₁ n₂, g₁ * g₂⟩

    Instances For
      source
      instance SemidirectProduct.instMulSemidirectProduct {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      Mul (N ⋊[φ] G)
      source
      theorem SemidirectProduct.mul_def {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (a : N ⋊[φ] G) (b : N ⋊[φ] G) :
      a * b = { left := a.left * ↑(↑φ a.right) b.left, right := a.right * b.right }
      source
      @[simp]
      theorem SemidirectProduct.mul_left {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (a : N ⋊[φ] G) (b : N ⋊[φ] G) :
      (a * b).left = a.left * ↑(↑φ a.right) b.left
      source
      @[simp]
      theorem SemidirectProduct.mul_right {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (a : N ⋊[φ] G) (b : N ⋊[φ] G) :
      (a * b).right = a.right * b.right
      source
      instance SemidirectProduct.instOneSemidirectProduct {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      One (N ⋊[φ] G)
      source
      @[simp]
      theorem SemidirectProduct.one_left {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      1.left = 1
      source
      @[simp]
      theorem SemidirectProduct.one_right {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      1.right = 1
      source
      instance SemidirectProduct.instInvSemidirectProduct {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      Inv (N ⋊[φ] G)
      source
      @[simp]
      theorem SemidirectProduct.inv_left {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (a : N ⋊[φ] G) :
      a⁻¹.left = ↑(↑φ a.right⁻¹) a.left⁻¹
      source
      @[simp]
      theorem SemidirectProduct.inv_right {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (a : N ⋊[φ] G) :
      a⁻¹.right = a.right⁻¹
      source
      instance SemidirectProduct.instGroupSemidirectProduct {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      Group (N ⋊[φ] G)
      source
      instance SemidirectProduct.instInhabitedSemidirectProduct {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      Inhabited (N ⋊[φ] G)
      source
      def SemidirectProduct.inl {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      N →* N ⋊[φ] G

      The canonical map N →* N ⋊[φ] G sending n to ⟨n, 1⟩

      Instances For
        source
        @[simp]
        theorem SemidirectProduct.left_inl {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (n : N) :
        (↑SemidirectProduct.inl n).left = n
        source
        @[simp]
        theorem SemidirectProduct.right_inl {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (n : N) :
        (↑SemidirectProduct.inl n).right = 1
        source
        theorem SemidirectProduct.inl_injective {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
        Function.Injective ↑SemidirectProduct.inl
        source
        @[simp]
        theorem SemidirectProduct.inl_inj {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {n₁ : N} {n₂ : N} :
        ↑SemidirectProduct.inl n₁ = ↑SemidirectProduct.inl n₂ ↔ n₁ = n₂
        source
        def SemidirectProduct.inr {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
        G →* N ⋊[φ] G

        The canonical map G →* N ⋊[φ] G sending g to ⟨1, g⟩

        Instances For
          source
          @[simp]
          theorem SemidirectProduct.left_inr {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (g : G) :
          (↑SemidirectProduct.inr g).left = 1
          source
          @[simp]
          theorem SemidirectProduct.right_inr {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (g : G) :
          (↑SemidirectProduct.inr g).right = g
          source
          theorem SemidirectProduct.inr_injective {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
          Function.Injective ↑SemidirectProduct.inr
          source
          @[simp]
          theorem SemidirectProduct.inr_inj {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {g₁ : G} {g₂ : G} :
          ↑SemidirectProduct.inr g₁ = ↑SemidirectProduct.inr g₂ ↔ g₁ = g₂
          source
          theorem SemidirectProduct.inl_aut {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (g : G) (n : N) :
          ↑SemidirectProduct.inl (↑(↑φ g) n) = ↑SemidirectProduct.inr g * ↑SemidirectProduct.inl n * ↑SemidirectProduct.inr g⁻¹
          source
          theorem SemidirectProduct.inl_aut_inv {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (g : G) (n : N) :
          ↑SemidirectProduct.inl (↑(↑φ g)⁻¹ n) = ↑SemidirectProduct.inr g⁻¹ * ↑SemidirectProduct.inl n * ↑SemidirectProduct.inr g
          source
          @[simp]
          theorem SemidirectProduct.mk_eq_inl_mul_inr {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (g : G) (n : N) :
          { left := n, right := g } = ↑SemidirectProduct.inl n * ↑SemidirectProduct.inr g
          source
          @[simp]
          theorem SemidirectProduct.inl_left_mul_inr_right {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (x : N ⋊[φ] G) :
          ↑SemidirectProduct.inl x.left * ↑SemidirectProduct.inr x.right = x
          source
          def SemidirectProduct.rightHom {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
          N ⋊[φ] G →* G

          The canonical projection map N ⋊[φ] G →* G, as a group hom.

          Instances For
            source
            @[simp]
            theorem SemidirectProduct.rightHom_eq_right {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
            ↑SemidirectProduct.rightHom = SemidirectProduct.right
            source
            @[simp]
            theorem SemidirectProduct.rightHom_comp_inl {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
            MonoidHom.comp SemidirectProduct.rightHom SemidirectProduct.inl = 1
            source
            @[simp]
            theorem SemidirectProduct.rightHom_comp_inr {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
            MonoidHom.comp SemidirectProduct.rightHom SemidirectProduct.inr = MonoidHom.id G
            source
            @[simp]
            theorem SemidirectProduct.rightHom_inl {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (n : N) :
            ↑SemidirectProduct.rightHom (↑SemidirectProduct.inl n) = 1
            source
            @[simp]
            theorem SemidirectProduct.rightHom_inr {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} (g : G) :
            ↑SemidirectProduct.rightHom (↑SemidirectProduct.inr g) = g
            source
            theorem SemidirectProduct.rightHom_surjective {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
            Function.Surjective ↑SemidirectProduct.rightHom
            source
            theorem SemidirectProduct.range_inl_eq_ker_rightHom {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
            MonoidHom.range SemidirectProduct.inl = MonoidHom.ker SemidirectProduct.rightHom
            source
            def SemidirectProduct.lift {N : Type u_1} {G : Type u_2} {H : Type u_3} [Group N] [Group G] [Group H] {φ : G →* MulAut N} (f₁ : N →* H) (f₂ : G →* H) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁) :
            N ⋊[φ] G →* H

            Define a group hom N ⋊[φ] G →* H, by defining maps N →* H and G →* H

            Instances For
              source
              @[simp]
              theorem SemidirectProduct.lift_inl {N : Type u_1} {G : Type u_2} {H : Type u_3} [Group N] [Group G] [Group H] {φ : G →* MulAut N} (f₁ : N →* H) (f₂ : G →* H) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁) (n : N) :
              ↑(SemidirectProduct.lift f₁ f₂ h) (↑SemidirectProduct.inl n) = ↑f₁ n
              source
              @[simp]
              theorem SemidirectProduct.lift_comp_inl {N : Type u_1} {G : Type u_2} {H : Type u_3} [Group N] [Group G] [Group H] {φ : G →* MulAut N} (f₁ : N →* H) (f₂ : G →* H) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁) :
              MonoidHom.comp (SemidirectProduct.lift f₁ f₂ h) SemidirectProduct.inl = f₁
              source
              @[simp]
              theorem SemidirectProduct.lift_inr {N : Type u_1} {G : Type u_2} {H : Type u_3} [Group N] [Group G] [Group H] {φ : G →* MulAut N} (f₁ : N →* H) (f₂ : G →* H) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁) (g : G) :
              ↑(SemidirectProduct.lift f₁ f₂ h) (↑SemidirectProduct.inr g) = ↑f₂ g
              source
              @[simp]
              theorem SemidirectProduct.lift_comp_inr {N : Type u_1} {G : Type u_2} {H : Type u_3} [Group N] [Group G] [Group H] {φ : G →* MulAut N} (f₁ : N →* H) (f₂ : G →* H) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁) :
              MonoidHom.comp (SemidirectProduct.lift f₁ f₂ h) SemidirectProduct.inr = f₂
              source
              theorem SemidirectProduct.lift_unique {N : Type u_1} {G : Type u_2} {H : Type u_3} [Group N] [Group G] [Group H] {φ : G →* MulAut N} (F : N ⋊[φ] G →* H) :
              F = SemidirectProduct.lift (MonoidHom.comp F SemidirectProduct.inl) (MonoidHom.comp F SemidirectProduct.inr) (_ : ∀ (x : G), MonoidHom.comp (MonoidHom.comp F SemidirectProduct.inl) (MulEquiv.toMonoidHom (↑φ x)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑(MonoidHom.comp F SemidirectProduct.inr) x))) (MonoidHom.comp F SemidirectProduct.inl))
              source
              theorem SemidirectProduct.hom_ext {N : Type u_1} {G : Type u_2} {H : Type u_3} [Group N] [Group G] [Group H] {φ : G →* MulAut N} {f : N ⋊[φ] G →* H} {g : N ⋊[φ] G →* H} (hl : MonoidHom.comp f SemidirectProduct.inl = MonoidHom.comp g SemidirectProduct.inl) (hr : MonoidHom.comp f SemidirectProduct.inr = MonoidHom.comp g SemidirectProduct.inr) :
              f = g

              Two maps out of the semidirect product are equal if they're equal after composition with both inl and inr

              source
              def SemidirectProduct.map {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {N₁ : Type u_4} {G₁ : Type u_5} [Group N₁] [Group G₁] {φ₁ : G₁ →* MulAut N₁} (f₁ : N →* N₁) (f₂ : G →* G₁) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁) :
              N ⋊[φ] G →* N₁ ⋊[φ₁] G₁

              Define a map from N ⋊[φ] G to N₁ ⋊[φ₁] G₁ given maps N →* N₁ and G →* G₁ that satisfy a commutativity condition ∀ n g, f₁ (φ g n) = φ₁ (f₂ g) (f₁ n).

              Instances For
                source
                @[simp]
                theorem SemidirectProduct.map_left {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {N₁ : Type u_4} {G₁ : Type u_5} [Group N₁] [Group G₁] {φ₁ : G₁ →* MulAut N₁} (f₁ : N →* N₁) (f₂ : G →* G₁) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁) (g : N ⋊[φ] G) :
                (↑(SemidirectProduct.map f₁ f₂ h) g).left = ↑f₁ g.left
                source
                @[simp]
                theorem SemidirectProduct.map_right {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {N₁ : Type u_4} {G₁ : Type u_5} [Group N₁] [Group G₁] {φ₁ : G₁ →* MulAut N₁} (f₁ : N →* N₁) (f₂ : G →* G₁) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁) (g : N ⋊[φ] G) :
                (↑(SemidirectProduct.map f₁ f₂ h) g).right = ↑f₂ g.right
                source
                @[simp]
                theorem SemidirectProduct.rightHom_comp_map {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {N₁ : Type u_4} {G₁ : Type u_5} [Group N₁] [Group G₁] {φ₁ : G₁ →* MulAut N₁} (f₁ : N →* N₁) (f₂ : G →* G₁) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁) :
                MonoidHom.comp SemidirectProduct.rightHom (SemidirectProduct.map f₁ f₂ h) = MonoidHom.comp f₂ SemidirectProduct.rightHom
                source
                @[simp]
                theorem SemidirectProduct.map_inl {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {N₁ : Type u_4} {G₁ : Type u_5} [Group N₁] [Group G₁] {φ₁ : G₁ →* MulAut N₁} (f₁ : N →* N₁) (f₂ : G →* G₁) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁) (n : N) :
                ↑(SemidirectProduct.map f₁ f₂ h) (↑SemidirectProduct.inl n) = ↑SemidirectProduct.inl (↑f₁ n)
                source
                @[simp]
                theorem SemidirectProduct.map_comp_inl {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {N₁ : Type u_4} {G₁ : Type u_5} [Group N₁] [Group G₁] {φ₁ : G₁ →* MulAut N₁} (f₁ : N →* N₁) (f₂ : G →* G₁) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁) :
                MonoidHom.comp (SemidirectProduct.map f₁ f₂ h) SemidirectProduct.inl = MonoidHom.comp SemidirectProduct.inl f₁
                source
                @[simp]
                theorem SemidirectProduct.map_inr {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {N₁ : Type u_4} {G₁ : Type u_5} [Group N₁] [Group G₁] {φ₁ : G₁ →* MulAut N₁} (f₁ : N →* N₁) (f₂ : G →* G₁) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁) (g : G) :
                ↑(SemidirectProduct.map f₁ f₂ h) (↑SemidirectProduct.inr g) = ↑SemidirectProduct.inr (↑f₂ g)
                source
                @[simp]
                theorem SemidirectProduct.map_comp_inr {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} {N₁ : Type u_4} {G₁ : Type u_5} [Group N₁] [Group G₁] {φ₁ : G₁ →* MulAut N₁} (f₁ : N →* N₁) (f₂ : G →* G₁) (h : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁) :
                MonoidHom.comp (SemidirectProduct.map f₁ f₂ h) SemidirectProduct.inr = MonoidHom.comp SemidirectProduct.inr f₂