p-groups

This file contains a proof that if G is a p-group acting on a finite set α, then the number of fixed points of the action is congruent mod p to the cardinality of α. It also contains proofs of some corollaries of this lemma about existence of fixed points.

def IsPGroup (p : ℕ) (G : Type u_1) [Group G] : Prop

A p-group is a group in which every element has prime power order

Equations

• IsPGroup p G = ∀ (g : G), ∃ (k : ℕ), g ^ p ^ k = 1

theorem IsPGroup.iff_orderOf {p : ℕ} {G : Type u_1} [Group G] [hp : Fact (Nat.Prime p)] : IsPGroup p G ↔ ∀ (g : G), ∃ (k : ℕ), orderOf g = p ^ k

theorem IsPGroup.of_card {p : ℕ} {G : Type u_1} [Group G] {n : ℕ} (hG : Nat.card G = p ^ n) : IsPGroup p G

theorem IsPGroup.of_bot {p : ℕ} {G : Type u_1} [Group G] : IsPGroup p { x : G // x ∈ ⊥ }

theorem IsPGroup.iff_card {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] [Finite G] : IsPGroup p G ↔ ∃ (n : ℕ), Nat.card G = p ^ n

theorem IsPGroup.exists_card_eq {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] [Finite G] : IsPGroup p G → ∃ (n : ℕ), Nat.card G = p ^ n

Alias of the forward direction of IsPGroup.iff_card.

theorem IsPGroup.of_injective {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) {H : Type u_2} [Group H] (ϕ : H →* G) (hϕ : Function.Injective ⇑ϕ) : IsPGroup p H

theorem IsPGroup.to_subgroup {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) (H : Subgroup G) : IsPGroup p { x : G // x ∈ H }

theorem IsPGroup.of_surjective {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) {H : Type u_2} [Group H] (ϕ : G →* H) (hϕ : Function.Surjective ⇑ϕ) : IsPGroup p H

theorem IsPGroup.to_quotient {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) (H : Subgroup G) [H.Normal] : IsPGroup p (G ⧸ H)

theorem IsPGroup.of_equiv {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) {H : Type u_2} [Group H] (ϕ : G ≃* H) : IsPGroup p H

theorem IsPGroup.orderOf_coprime {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) {n : ℕ} (hn : p.Coprime n) (g : G) : (orderOf g).Coprime n

noncomputable def IsPGroup.powEquiv {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) {n : ℕ} (hn : p.Coprime n) : G ≃ G

If gcd(p,n) = 1, then the nth power map is a bijection.

Equations

• hG.powEquiv hn = { toFun := fun (x : G) => x ^ n, invFun := fun (g : G) => ↑((powCoprime ⋯).symm ⟨g, ⋯⟩), left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem IsPGroup.powEquiv_apply {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) {n : ℕ} (hn : p.Coprime n) (g : G) : (hG.powEquiv hn) g = g ^ n

@[simp]

theorem IsPGroup.powEquiv_symm_apply {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) {n : ℕ} (hn : p.Coprime n) (g : G) : (hG.powEquiv hn).symm g = g ^ (orderOf g).gcdB n

@[reducible, inline]

noncomputable abbrev IsPGroup.powEquiv' {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] {n : ℕ} (hn : ¬p ∣ n) : G ≃ G

If p ∤ n, then the nth power map is a bijection.

Equations

• hG.powEquiv' hn = hG.powEquiv ⋯

theorem IsPGroup.index {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] (H : Subgroup G) [H.FiniteIndex] : ∃ (n : ℕ), H.index = p ^ n

theorem IsPGroup.card_eq_or_dvd {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] : Nat.card G = 1 ∨ p ∣ Nat.card G

theorem IsPGroup.nontrivial_iff_card {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] [Finite G] : Nontrivial G ↔ ∃ n > 0, Nat.card G = p ^ n

theorem IsPGroup.card_orbit {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] {α : Type u_2} [MulAction G α] (a : α) [Finite ↑(MulAction.orbit G a)] : ∃ (n : ℕ), Nat.card ↑(MulAction.orbit G a) = p ^ n

theorem IsPGroup.card_modEq_card_fixedPoints {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] (α : Type u_2) [MulAction G α] [Finite α] : Nat.card α ≡ Nat.card ↑(MulAction.fixedPoints G α) [MOD p]

If G is a p-group acting on a finite set α, then the number of fixed points of the action is congruent mod p to the cardinality of α

theorem IsPGroup.nonempty_fixed_point_of_prime_not_dvd_card {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] (α : Type u_3) [MulAction G α] (hpα : ¬p ∣ Nat.card α) : (MulAction.fixedPoints G α).Nonempty

If a p-group acts on α and the cardinality of α is not a multiple of p then the action has a fixed point.

theorem IsPGroup.exists_fixed_point_of_prime_dvd_card_of_fixed_point {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] (α : Type u_2) [MulAction G α] [Finite α] (hpα : p ∣ Nat.card α) {a : α} (ha : a ∈ MulAction.fixedPoints G α) : ∃ b ∈ MulAction.fixedPoints G α, a ≠ b

If a p-group acts on α and the cardinality of α is a multiple of p, and the action has one fixed point, then it has another fixed point.

theorem IsPGroup.center_nontrivial {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] [Nontrivial G] [Finite G] : Nontrivial { x : G // x ∈ Subgroup.center G }

theorem IsPGroup.bot_lt_center {p : ℕ} {G : Type u_1} [Group G] (hG : IsPGroup p G) [hp : Fact (Nat.Prime p)] [Nontrivial G] [Finite G] : ⊥ < Subgroup.center G

theorem IsPGroup.to_le {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (hK : IsPGroup p { x : G // x ∈ K }) (hHK : H ≤ K) : IsPGroup p { x : G // x ∈ H }

theorem IsPGroup.to_inf_left {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (hH : IsPGroup p { x : G // x ∈ H }) : IsPGroup p { x : G // x ∈ H ⊓ K }

theorem IsPGroup.to_inf_right {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (hK : IsPGroup p { x : G // x ∈ K }) : IsPGroup p { x : G // x ∈ H ⊓ K }

theorem IsPGroup.map {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} (hH : IsPGroup p { x : G // x ∈ H }) {K : Type u_2} [Group K] (ϕ : G →* K) : IsPGroup p { x : K // x ∈ Subgroup.map ϕ H }

theorem IsPGroup.comap_of_ker_isPGroup {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} (hH : IsPGroup p { x : G // x ∈ H }) {K : Type u_2} [Group K] (ϕ : K →* G) (hϕ : IsPGroup p { x : K // x ∈ ϕ.ker }) : IsPGroup p { x : K // x ∈ Subgroup.comap ϕ H }

theorem IsPGroup.ker_isPGroup_of_injective {p : ℕ} {G : Type u_1} [Group G] {K : Type u_2} [Group K] {ϕ : K →* G} (hϕ : Function.Injective ⇑ϕ) : IsPGroup p { x : K // x ∈ ϕ.ker }

theorem IsPGroup.comap_of_injective {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} (hH : IsPGroup p { x : G // x ∈ H }) {K : Type u_2} [Group K] (ϕ : K →* G) (hϕ : Function.Injective ⇑ϕ) : IsPGroup p { x : K // x ∈ Subgroup.comap ϕ H }

theorem IsPGroup.comap_subtype {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} (hH : IsPGroup p { x : G // x ∈ H }) {K : Subgroup G} : IsPGroup p { x : { x : G // x ∈ K } // x ∈ Subgroup.comap K.subtype H }

theorem IsPGroup.to_sup_of_normal_right {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (hH : IsPGroup p { x : G // x ∈ H }) (hK : IsPGroup p { x : G // x ∈ K }) [K.Normal] : IsPGroup p { x : G // x ∈ H ⊔ K }

theorem IsPGroup.to_sup_of_normal_left {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (hH : IsPGroup p { x : G // x ∈ H }) (hK : IsPGroup p { x : G // x ∈ K }) [H.Normal] : IsPGroup p { x : G // x ∈ H ⊔ K }

theorem IsPGroup.to_sup_of_normal_right' {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (hH : IsPGroup p { x : G // x ∈ H }) (hK : IsPGroup p { x : G // x ∈ K }) (hHK : H ≤ K.normalizer) : IsPGroup p { x : G // x ∈ H ⊔ K }

theorem IsPGroup.to_sup_of_normal_left' {p : ℕ} {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (hH : IsPGroup p { x : G // x ∈ H }) (hK : IsPGroup p { x : G // x ∈ K }) (hHK : K ≤ H.normalizer) : IsPGroup p { x : G // x ∈ H ⊔ K }

theorem IsPGroup.coprime_card_of_ne {G : Type u_1} [Group G] {G₂ : Type u_2} [Group G₂] (p₁ : ℕ) (p₂ : ℕ) [hp₁ : Fact (Nat.Prime p₁)] [hp₂ : Fact (Nat.Prime p₂)] (hne : p₁ ≠ p₂) (H₁ : Subgroup G) (H₂ : Subgroup G₂) [Finite { x : G // x ∈ H₁ }] [Finite { x : G₂ // x ∈ H₂ }] (hH₁ : IsPGroup p₁ { x : G // x ∈ H₁ }) (hH₂ : IsPGroup p₂ { x : G₂ // x ∈ H₂ }) : (Nat.card { x : G // x ∈ H₁ }).Coprime (Nat.card { x : G₂ // x ∈ H₂ })

finite p-groups with different p have coprime orders

theorem IsPGroup.disjoint_of_ne {G : Type u_1} [Group G] (p₁ : ℕ) (p₂ : ℕ) [hp₁ : Fact (Nat.Prime p₁)] [hp₂ : Fact (Nat.Prime p₂)] (hne : p₁ ≠ p₂) (H₁ : Subgroup G) (H₂ : Subgroup G) (hH₁ : IsPGroup p₁ { x : G // x ∈ H₁ }) (hH₂ : IsPGroup p₂ { x : G // x ∈ H₂ }) : Disjoint H₁ H₂

p-groups with different p are disjoint

theorem IsPGroup.card_center_eq_prime_pow {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] {n : ℕ} (hGpn : Nat.card G = p ^ n) (hn : 0 < n) : ∃ k > 0, Nat.card { x : G // x ∈ Subgroup.center G } = p ^ k

The cardinality of the center of a p-group is p ^ k where k is positive.

theorem IsPGroup.cyclic_center_quotient_of_card_eq_prime_sq {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] (hG : Nat.card G = p ^ 2) : IsCyclic (G ⧸ Subgroup.center G)

The quotient by the center of a group of cardinality p ^ 2 is cyclic.

def IsPGroup.commGroupOfCardEqPrimeSq {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] (hG : Nat.card G = p ^ 2) : CommGroup G

A group of order p ^ 2 is commutative. See also IsPGroup.commutative_of_card_eq_prime_sq for just the proof that ∀ a b, a * b = b * a

Equations

• IsPGroup.commGroupOfCardEqPrimeSq hG = commGroupOfCyclicCenterQuotient (QuotientGroup.mk' (Subgroup.center G)) ⋯

theorem IsPGroup.commutative_of_card_eq_prime_sq {p : ℕ} {G : Type u_1} [Group G] [Fact (Nat.Prime p)] (hG : Nat.card G = p ^ 2) (a : G) (b : G) : a * b = b * a

A group of order p ^ 2 is commutative. See also IsPGroup.commGroupOfCardEqPrimeSq for the CommGroup instance.