Commutators of Subgroups

If G is a group and H₁ H₂ : Subgroup G then the commutator ⁅H₁, H₂⁆ : Subgroup G is the subgroup of G generated by the commutators h₁ * h₂ * h₁⁻¹ * h₂⁻¹.

Main definitions

• ⁅g₁, g₂⁆ : the commutator of the elements g₁ and g₂ (defined by commutatorElement elsewhere).
• ⁅H₁, H₂⁆ : the commutator of the subgroups H₁ and H₂.
• commutator: defines the commutator of a group G as a subgroup of G.

theorem commutatorElement_eq_one_iff_mul_comm {G : Type u_1} [Group G] {g₁ g₂ : G} : ⁅g₁, g₂⁆ = 1 ↔ g₁ * g₂ = g₂ * g₁

theorem commutatorElement_eq_one_iff_commute {G : Type u_1} [Group G] {g₁ g₂ : G} : ⁅g₁, g₂⁆ = 1 ↔ Commute g₁ g₂

theorem Commute.commutator_eq {G : Type u_1} [Group G] {g₁ g₂ : G} (h : Commute g₁ g₂) : ⁅g₁, g₂⁆ = 1

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theorem commutatorElement_one_right {G : Type u_1} [Group G] (g : G) : ⁅g, 1⁆ = 1

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theorem commutatorElement_one_left {G : Type u_1} [Group G] (g : G) : ⁅1, g⁆ = 1

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theorem commutatorElement_self {G : Type u_1} [Group G] (g : G) : ⁅g, g⁆ = 1

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theorem commutatorElement_inv {G : Type u_1} [Group G] (g₁ g₂ : G) : ⁅g₁, g₂⁆⁻¹ = ⁅g₂, g₁⁆

theorem map_commutatorElement {G : Type u_1} {G' : Type u_2} {F : Type u_3} [Group G] [Group G'] [FunLike F G G'] [MonoidHomClass F G G'] (f : F) (g₁ g₂ : G) : f ⁅g₁, g₂⁆ = ⁅f g₁, f g₂⁆

theorem conjugate_commutatorElement {G : Type u_1} [Group G] (g₁ g₂ g₃ : G) : g₃ * ⁅g₁, g₂⁆ * g₃⁻¹ = ⁅g₃ * g₁ * g₃⁻¹, g₃ * g₂ * g₃⁻¹⁆

instance Subgroup.commutator {G : Type u_1} [Group G] : Bracket (Subgroup G) (Subgroup G)

The commutator of two subgroups H₁ and H₂.

Equations

• Subgroup.commutator = { bracket := fun (H₁ H₂ : Subgroup G) => Subgroup.closure {g : G | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = g} }

theorem Subgroup.commutator_def {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) : ⁅H₁, H₂⁆ = closure {g : G | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = g}

theorem Subgroup.commutator_mem_commutator {G : Type u_1} [Group G] {g₁ g₂ : G} {H₁ H₂ : Subgroup G} (h₁ : g₁ ∈ H₁) (h₂ : g₂ ∈ H₂) : ⁅g₁, g₂⁆ ∈ ⁅H₁, H₂⁆

theorem Subgroup.commutator_le {G : Type u_1} [Group G] {H₁ H₂ H₃ : Subgroup G} : ⁅H₁, H₂⁆ ≤ H₃ ↔ ∀ g₁ ∈ H₁, ∀ g₂ ∈ H₂, ⁅g₁, g₂⁆ ∈ H₃

theorem Subgroup.commutator_mono {G : Type u_1} [Group G] {H₁ H₂ K₁ K₂ : Subgroup G} (h₁ : H₁ ≤ K₁) (h₂ : H₂ ≤ K₂) : ⁅H₁, H₂⁆ ≤ ⁅K₁, K₂⁆

theorem Subgroup.commutator_eq_bot_iff_le_centralizer {G : Type u_1} [Group G] {H₁ H₂ : Subgroup G} : ⁅H₁, H₂⁆ = ⊥ ↔ H₁ ≤ centralizer ↑H₂

theorem Subgroup.commutator_commutator_eq_bot_of_rotate {G : Type u_1} [Group G] {H₁ H₂ H₃ : Subgroup G} (h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥) (h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥) : ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥

The Three Subgroups Lemma (via the Hall-Witt identity)

theorem Subgroup.commutator_comm_le {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) : ⁅H₁, H₂⁆ ≤ ⁅H₂, H₁⁆

theorem Subgroup.commutator_comm {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) : ⁅H₁, H₂⁆ = ⁅H₂, H₁⁆

instance Subgroup.commutator_normal {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) [h₁ : H₁.Normal] [h₂ : H₂.Normal] : ⁅H₁, H₂⁆.Normal

theorem Subgroup.commutator_def' {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) [H₁.Normal] [H₂.Normal] : ⁅H₁, H₂⁆ = normalClosure {g : G | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = g}

theorem Subgroup.commutator_le_right {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) [h : H₂.Normal] : ⁅H₁, H₂⁆ ≤ H₂

theorem Subgroup.commutator_le_left {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) [H₁.Normal] : ⁅H₁, H₂⁆ ≤ H₁

@[simp]

theorem Subgroup.commutator_bot_left {G : Type u_1} [Group G] (H₁ : Subgroup G) : ⁅⊥, H₁⁆ = ⊥

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theorem Subgroup.commutator_bot_right {G : Type u_1} [Group G] (H₁ : Subgroup G) : ⁅H₁, ⊥⁆ = ⊥

theorem Subgroup.commutator_le_inf {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) [H₁.Normal] [H₂.Normal] : ⁅H₁, H₂⁆ ≤ H₁ ⊓ H₂

theorem Subgroup.map_commutator {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H₁ H₂ : Subgroup G) (f : G →* G') : map f ⁅H₁, H₂⁆ = ⁅map f H₁, map f H₂⁆

theorem Subgroup.commutator_le_map_commutator {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] {H₁ H₂ : Subgroup G} {f : G →* G'} {K₁ K₂ : Subgroup G'} (h₁ : K₁ ≤ map f H₁) (h₂ : K₂ ≤ map f H₂) : ⁅K₁, K₂⁆ ≤ map f ⁅H₁, H₂⁆

instance Subgroup.commutator_characteristic {G : Type u_1} [Group G] (H₁ H₂ : Subgroup G) [h₁ : H₁.Characteristic] [h₂ : H₂.Characteristic] : ⁅H₁, H₂⁆.Characteristic

theorem Subgroup.commutator_prod_prod {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H₁ H₂ : Subgroup G) (K₁ K₂ : Subgroup G') : ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆

theorem Subgroup.commutator_pi_pi_le {η : Type u_4} {Gs : η → Type u_5} [(i : η) → Group (Gs i)] (H K : (i : η) → Subgroup (Gs i)) : ⁅pi Set.univ H, pi Set.univ K⁆ ≤ pi Set.univ fun (i : η) => ⁅H i, K i⁆

The commutator of direct product is contained in the direct product of the commutators.

See commutator_pi_pi_of_finite for equality given Fintype η.

def commutatorSet (G : Type u_1) [Group G] : Set G

The set of commutator elements ⁅g₁, g₂⁆ in G.

Equations

• commutatorSet G = {g : G | ∃ (g₁ : G) (g₂ : G), ⁅g₁, g₂⁆ = g}

theorem commutatorSet_def (G : Type u_1) [Group G] : commutatorSet G = {g : G | ∃ (g₁ : G) (g₂ : G), ⁅g₁, g₂⁆ = g}

theorem one_mem_commutatorSet (G : Type u_1) [Group G] : 1 ∈ commutatorSet G

instance instNonemptyElemCommutatorSet (G : Type u_1) [Group G] : Nonempty ↑(commutatorSet G)

theorem mem_commutatorSet_iff {G : Type u_1} [Group G] {g : G} : g ∈ commutatorSet G ↔ ∃ (g₁ : G) (g₂ : G), ⁅g₁, g₂⁆ = g

theorem commutator_mem_commutatorSet {G : Type u_1} [Group G] (g₁ g₂ : G) : ⁅g₁, g₂⁆ ∈ commutatorSet G

def commutator (G : Type u_1) [Group G] : Subgroup G

The commutator subgroup of a group G is the normal subgroup generated by the commutators [p,q]=p*q*p⁻¹*q⁻¹.

Equations

• commutator G = ⁅⊤, ⊤⁆

instance instNormalCommutator (G : Type u_1) [Group G] : (commutator G).Normal

theorem commutator_def (G : Type u_1) [Group G] : commutator G = ⁅⊤, ⊤⁆

theorem commutator_eq_closure (G : Type u_1) [Group G] : commutator G = Subgroup.closure (commutatorSet G)

theorem commutator_eq_normalClosure (G : Type u_1) [Group G] : commutator G = Subgroup.normalClosure (commutatorSet G)

theorem Subgroup.map_subtype_commutator {G : Type u_1} [Group G] (H : Subgroup G) : map H.subtype (_root_.commutator ↥H) = ⁅H, H⁆

theorem Subgroup.commutator_le_self {G : Type u_1} [Group G] (H : Subgroup G) : ⁅H, H⁆ ≤ H

instance commutator_characteristic (G : Type u_1) [Group G] : (commutator G).Characteristic

theorem commutator_eq_bot_iff_center_eq_top (G : Type u_1) [Group G] : commutator G = ⊥ ↔ Subgroup.center G = ⊤

theorem commutator_centralizer_commutator_le_center (G : Type u_1) [Group G] : ⁅Subgroup.centralizer ↑(commutator G), Subgroup.centralizer ↑(commutator G)⁆ ≤ Subgroup.center G

theorem mem_commutatorSet_of_isConj_sq (G : Type u_1) [Group G] {g : G} (hg : IsConj g (g ^ 2)) : g ∈ commutatorSet G

If g is conjugate to g ^ 2, then g is a commutator

theorem map_commutator_eq (G : Type u_1) [Group G] {H : Type u_4} [Group H] (f : G →* H) : Subgroup.map f (commutator G) = ⁅f.range, f.range⁆

def commutatorRepresentatives (G : Type u_1) [Group G] : Set (G × G)

Representatives (g₁, g₂) : G × G of commutators ⁅g₁, g₂⁆ ∈ G.

Equations

• commutatorRepresentatives G = Set.range fun (g : ↑(commutatorSet G)) => (Exists.choose ⋯, ⋯.choose)

def closureCommutatorRepresentatives (G : Type u_1) [Group G] : Subgroup G

Subgroup generated by representatives g₁ g₂ : G of commutators ⁅g₁, g₂⁆ ∈ G.

Equations

• closureCommutatorRepresentatives G = Subgroup.closure (Prod.fst '' commutatorRepresentatives G ∪ Prod.snd '' commutatorRepresentatives G)

theorem image_commutatorSet_closureCommutatorRepresentatives (G : Type u_1) [Group G] : ⇑(closureCommutatorRepresentatives G).subtype '' commutatorSet ↥(closureCommutatorRepresentatives G) = commutatorSet G

theorem Subgroup.Normal.quotient_commutative_iff_commutator_le {G : Type u_1} [Group G] {N : Subgroup G} [N.Normal] : (Std.Commutative fun (x1 x2 : G ⧸ N) => x1 * x2) ↔ _root_.commutator G ≤ N

theorem Subgroup.Normal.commutator_le_of_self_sup_commutative_eq_top {G : Type u_1} [Group G] {N : Subgroup G} [N.Normal] {H : Subgroup G} (hHN : N ⊔ H = ⊤) (hH : IsMulCommutative ↥H) : _root_.commutator G ≤ N

If N is a normal subgroup of G and H a commutative subgroup such that H ⊔ N = ⊤, then N contains commutator G.