# mathlibdocumentation

group_theory.p_group

# 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 is_p_group (p : ) (G : Type u_1) [group G] :
Prop

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

Equations
• G = ∀ (g : G), ∃ (k : ), g ^ p ^ k = 1
theorem is_p_group.iff_order_of {p : } {G : Type u_1} [group G] [hp : fact (nat.prime p)] :
G ∀ (g : G), ∃ (k : ), = p ^ k
theorem is_p_group.of_card {p : } {G : Type u_1} [group G] [fintype G] {n : } (hG : = p ^ n) :
G
theorem is_p_group.of_bot {p : } {G : Type u_1} [group G] :
theorem is_p_group.iff_card {p : } {G : Type u_1} [group G] [fact (nat.prime p)] [fintype G] :
G ∃ (n : ), = p ^ n
theorem is_p_group.of_injective {p : } {G : Type u_1} [group G] (hG : G) {H : Type u_2} [group H] (ϕ : H →* G) (hϕ : function.injective ϕ) :
H
theorem is_p_group.to_subgroup {p : } {G : Type u_1} [group G] (hG : G) (H : subgroup G) :
H
theorem is_p_group.of_surjective {p : } {G : Type u_1} [group G] (hG : G) {H : Type u_2} [group H] (ϕ : G →* H) (hϕ : function.surjective ϕ) :
H
theorem is_p_group.to_quotient {p : } {G : Type u_1} [group G] (hG : G) (H : subgroup G) [H.normal] :
(G H)
theorem is_p_group.of_equiv {p : } {G : Type u_1} [group G] (hG : G) {H : Type u_2} [group H] (ϕ : G ≃* H) :
H
theorem is_p_group.index {p : } {G : Type u_1} [group G] (hG : G) [hp : fact (nat.prime p)] (H : subgroup G) [finite (G H)] :
∃ (n : ), H.index = p ^ n
theorem is_p_group.nontrivial_iff_card {p : } {G : Type u_1} [group G] (hG : G) [hp : fact (nat.prime p)] [fintype G] :
∃ (n : ) (H : n > 0), = p ^ n
theorem is_p_group.card_orbit {p : } {G : Type u_1} [group G] (hG : G) [hp : fact (nat.prime p)] {α : Type u_2} [ α] (a : α) [fintype a)] :
∃ (n : ), = p ^ n
theorem is_p_group.card_modeq_card_fixed_points {p : } {G : Type u_1} [group G] (hG : G) [hp : fact (nat.prime p)] (α : Type u_2) [ α] [fintype α] [fintype ] :

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 is_p_group.nonempty_fixed_point_of_prime_not_dvd_card {p : } {G : Type u_1} [group G] (hG : G) [hp : fact (nat.prime p)] (α : Type u_2) [ α] [fintype α] (hpα : ¬) [finite ] :

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 is_p_group.exists_fixed_point_of_prime_dvd_card_of_fixed_point {p : } {G : Type u_1} [group G] (hG : G) [hp : fact (nat.prime p)] (α : Type u_2) [ α] [fintype α] (hpα : p ) {a : α} (ha : a ) :
∃ (b : 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 is_p_group.center_nontrivial {p : } {G : Type u_1} [group G] (hG : G) [hp : fact (nat.prime p)] [nontrivial G] [finite G] :
theorem is_p_group.bot_lt_center {p : } {G : Type u_1} [group G] (hG : G) [hp : fact (nat.prime p)] [nontrivial G] [finite G] :
theorem is_p_group.to_le {p : } {G : Type u_1} [group G] {H K : subgroup G} (hK : K) (hHK : H K) :
H
theorem is_p_group.to_inf_left {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : H) :
(H K)
theorem is_p_group.to_inf_right {p : } {G : Type u_1} [group G] {H K : subgroup G} (hK : K) :
(H K)
theorem is_p_group.map {p : } {G : Type u_1} [group G] {H : subgroup G} (hH : H) {K : Type u_2} [group K] (ϕ : G →* K) :
H)
theorem is_p_group.comap_of_ker_is_p_group {p : } {G : Type u_1} [group G] {H : subgroup G} (hH : H) {K : Type u_2} [group K] (ϕ : K →* G) (hϕ : (ϕ.ker)) :
H)
theorem is_p_group.ker_is_p_group_of_injective {p : } {G : Type u_1} [group G] {K : Type u_2} [group K] {ϕ : K →* G} (hϕ : function.injective ϕ) :
(ϕ.ker)
theorem is_p_group.comap_of_injective {p : } {G : Type u_1} [group G] {H : subgroup G} (hH : H) {K : Type u_2} [group K] (ϕ : K →* G) (hϕ : function.injective ϕ) :
H)
theorem is_p_group.comap_subtype {p : } {G : Type u_1} [group G] {H : subgroup G} (hH : H) {K : subgroup G} :
H)
theorem is_p_group.to_sup_of_normal_right {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : H) (hK : K) [K.normal] :
(H K)
theorem is_p_group.to_sup_of_normal_left {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : H) (hK : K) [H.normal] :
(H K)
theorem is_p_group.to_sup_of_normal_right' {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : H) (hK : K) (hHK : H K.normalizer) :
(H K)
theorem is_p_group.to_sup_of_normal_left' {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : H) (hK : K) (hHK : K H.normalizer) :
(H K)
theorem is_p_group.coprime_card_of_ne {G : Type u_1} [group G] {G₂ : Type u_2} [group G₂] (p₁ p₂ : ) [hp₁ : fact (nat.prime p₁)] [hp₂ : fact (nat.prime p₂)] (hne : p₁ p₂) (H₁ : subgroup G) (H₂ : subgroup G₂) [fintype H₁] [fintype H₂] (hH₁ : H₁) (hH₂ : H₂) :

finite p-groups with different p have coprime orders

theorem is_p_group.disjoint_of_ne {G : Type u_1} [group G] (p₁ p₂ : ) [hp₁ : fact (nat.prime p₁)] [hp₂ : fact (nat.prime p₂)] (hne : p₁ p₂) (H₁ H₂ : subgroup G) (hH₁ : H₁) (hH₂ : H₂) :
disjoint H₁ H₂

p-groups with different p are disjoint

theorem is_p_group.card_center_eq_prime_pow {p : } {G : Type u_1} [group G] [fintype G] [fact (nat.prime p)] {n : } (hGpn : = p ^ n) (hn : 0 < n) [fintype ] :
∃ (k : ) (H : k > 0), = p ^ k

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

theorem is_p_group.cyclic_center_quotient_of_card_eq_prime_sq {p : } {G : Type u_1} [group G] [fintype G] [fact (nat.prime p)] (hG : = p ^ 2) :

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

def is_p_group.comm_group_of_card_eq_prime_sq {p : } {G : Type u_1} [group G] [fintype G] [fact (nat.prime p)] (hG : = p ^ 2) :

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

Equations
• = is_p_group.comm_group_of_card_eq_prime_sq._proof_3
theorem is_p_group.commutative_of_card_eq_prime_sq {p : } {G : Type u_1} [group G] [fintype G] [fact (nat.prime p)] (hG : = p ^ 2) (a b : G) :
a * b = b * a

A group of order p ^ 2 is commutative. See also is_p_group.comm_group_of_card_eq_prime_sq for the comm_group instance.