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
structure SemidirectProduct (N : Type u_1) (G : Type u_2) [Group N] [Group G] (φ : G →* MulAut N) :
Type (max u_1 u_2)

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⟩

  • left : N

    The element of N

  • right : G

    The element of G

Instances For
    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
    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)
    Equations
    • instDecidableEqSemidirectProduct = decEqSemidirectProduct✝
    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⟩

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      source
      instance SemidirectProduct.instMul {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      Mul (N ⋊[φ] G)
      Equations
      • SemidirectProduct.instMul = { mul := fun (a b : N ⋊[φ] G) => { left := a.left * (φ a.right) b.left, right := a.right * b.right } }
      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.instOne {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      One (N ⋊[φ] G)
      Equations
      • SemidirectProduct.instOne = { one := { left := 1, right := 1 } }
      source
      @[simp]
      theorem SemidirectProduct.one_left {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      SemidirectProduct.left 1 = 1
      source
      @[simp]
      theorem SemidirectProduct.one_right {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      SemidirectProduct.right 1 = 1
      source
      instance SemidirectProduct.instInv {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      Inv (N ⋊[φ] G)
      Equations
      • SemidirectProduct.instInv = { inv := fun (x : N ⋊[φ] G) => { left := (φ x.right⁻¹) x.left⁻¹, right := x.right⁻¹ } }
      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.instGroup {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      Group (N ⋊[φ] G)
      Equations
      source
      instance SemidirectProduct.instInhabited {N : Type u_1} {G : Type u_2} [Group N] [Group G] {φ : G →* MulAut N} :
      Inhabited (N ⋊[φ] G)
      Equations
      • SemidirectProduct.instInhabited = { default := 1 }
      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⟩

      Equations
      • SemidirectProduct.inl = { toFun := fun (n : N) => { left := n, right := 1 }, map_one' := ⋯, map_mul' := ⋯ }
      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⟩

        Equations
        • SemidirectProduct.inr = { toFun := fun (g : G) => { left := 1, right := g }, map_one' := ⋯, map_mul' := ⋯ }
        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.

          Equations
          • SemidirectProduct.rightHom = { toFun := SemidirectProduct.right, map_one' := ⋯, map_mul' := ⋯ }
          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} :
            SemidirectProduct.rightHom.comp 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} :
            SemidirectProduct.rightHom.comp 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} :
            SemidirectProduct.inl.range = SemidirectProduct.rightHom.ker
            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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (f₂ g))).comp f₁) :
            N ⋊[φ] G →* H

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

            Equations
            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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (f₂ g))).comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (f₂ g))).comp f₁) :
              (SemidirectProduct.lift f₁ f₂ h).comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (f₂ g))).comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (f₂ g))).comp f₁) :
              (SemidirectProduct.lift f₁ f₂ h).comp 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 (F.comp SemidirectProduct.inl) (F.comp SemidirectProduct.inr) ⋯
              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 : f.comp SemidirectProduct.inl = g.comp SemidirectProduct.inl) (hr : f.comp SemidirectProduct.inr = g.comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (φ₁ (f₂ g))).comp 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).

              Equations
              • SemidirectProduct.map f₁ f₂ h = { toFun := fun (x : N ⋊[φ] G) => { left := f₁ x.left, right := f₂ x.right }, map_one' := ⋯, map_mul' := ⋯ }
              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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (φ₁ (f₂ g))).comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (φ₁ (f₂ g))).comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (φ₁ (f₂ g))).comp f₁) :
                SemidirectProduct.rightHom.comp (SemidirectProduct.map f₁ f₂ h) = f₂.comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (φ₁ (f₂ g))).comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (φ₁ (f₂ g))).comp f₁) :
                (SemidirectProduct.map f₁ f₂ h).comp SemidirectProduct.inl = SemidirectProduct.inl.comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (φ₁ (f₂ g))).comp 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), f₁.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (φ₁ (f₂ g))).comp f₁) :
                (SemidirectProduct.map f₁ f₂ h).comp SemidirectProduct.inr = SemidirectProduct.inr.comp f₂