mathlib documentation

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
theorem is_p_group.iff_order_of {p : } {G : Type u_1} [group G] [hp : fact (nat.prime p)] :
is_p_group p G ∀ (g : G), ∃ (k : ), order_of g = p ^ k
theorem is_p_group.of_card {p : } {G : Type u_1} [group G] [fintype G] {n : } (hG : fintype.card G = p ^ n) :
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] :
is_p_group p G ∃ (n : ), fintype.card G = p ^ n
theorem is_p_group.of_injective {p : } {G : Type u_1} [group G] (hG : is_p_group p G) {H : Type u_2} [group H] (ϕ : H →* G) (hϕ : function.injective ϕ) :
theorem is_p_group.to_subgroup {p : } {G : Type u_1} [group G] (hG : is_p_group p G) (H : subgroup G) :
theorem is_p_group.of_surjective {p : } {G : Type u_1} [group G] (hG : is_p_group p G) {H : Type u_2} [group H] (ϕ : G →* H) (hϕ : function.surjective ϕ) :
theorem is_p_group.to_quotient {p : } {G : Type u_1} [group G] (hG : is_p_group p G) (H : subgroup G) [H.normal] :
theorem is_p_group.of_equiv {p : } {G : Type u_1} [group G] (hG : is_p_group p G) {H : Type u_2} [group H] (ϕ : G ≃* H) :
theorem is_p_group.index {p : } {G : Type u_1} [group G] (hG : is_p_group p G) [hp : fact (nat.prime p)] (H : subgroup G) [fintype (quotient_group.quotient H)] :
∃ (n : ), H.index = p ^ n
theorem is_p_group.card_orbit {p : } {G : Type u_1} [group G] (hG : is_p_group p G) [hp : fact (nat.prime p)] {α : Type u_2} [mul_action G α] (a : α) [fintype (mul_action.orbit G a)] :
∃ (n : ), fintype.card (mul_action.orbit G a) = p ^ n
theorem is_p_group.card_modeq_card_fixed_points {p : } {G : Type u_1} [group G] (hG : is_p_group p G) [hp : fact (nat.prime p)] (α : Type u_2) [mul_action G α] [fintype α] [fintype (mul_action.fixed_points G α)] :

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 : is_p_group p G) [hp : fact (nat.prime p)] (α : Type u_2) [mul_action G α] [fintype α] [fintype (mul_action.fixed_points G α)] (hpα : ¬p fintype.card α) :

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 : is_p_group p G) [hp : fact (nat.prime p)] (α : Type u_2) [mul_action G α] [fintype α] [fintype (mul_action.fixed_points G α)] (hpα : p fintype.card α) {a : α} (ha : a mul_action.fixed_points G α) :
∃ (b : α), b mul_action.fixed_points 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 is_p_group.to_le {p : } {G : Type u_1} [group G] {H K : subgroup G} (hK : is_p_group p K) (hHK : H K) :
theorem is_p_group.to_inf_left {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : is_p_group p H) :
theorem is_p_group.to_inf_right {p : } {G : Type u_1} [group G] {H K : subgroup G} (hK : is_p_group p K) :
theorem is_p_group.map {p : } {G : Type u_1} [group G] {H : subgroup G} (hH : is_p_group p H) {K : Type u_2} [group K] (ϕ : G →* K) :
theorem is_p_group.comap_of_ker_is_p_group {p : } {G : Type u_1} [group G] {H : subgroup G} (hH : is_p_group p H) {K : Type u_2} [group K] (ϕ : K →* G) (hϕ : is_p_group p (ϕ.ker)) :
theorem is_p_group.comap_of_injective {p : } {G : Type u_1} [group G] {H : subgroup G} (hH : is_p_group p H) {K : Type u_2} [group K] (ϕ : K →* G) (hϕ : function.injective ϕ) :
theorem is_p_group.to_sup_of_normal_right {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : is_p_group p H) (hK : is_p_group p K) [K.normal] :
theorem is_p_group.to_sup_of_normal_left {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : is_p_group p H) (hK : is_p_group p K) [H.normal] :
theorem is_p_group.to_sup_of_normal_right' {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : is_p_group p H) (hK : is_p_group p K) (hHK : H K.normalizer) :
theorem is_p_group.to_sup_of_normal_left' {p : } {G : Type u_1} [group G] {H K : subgroup G} (hH : is_p_group p H) (hK : is_p_group p K) (hHK : K H.normalizer) :