Section Frattini3.
Variables (gT : finGroupType) (p : nat) (P : {group gT}).
Hypothesis pP : p.-group P.

(* Specifications: *)
Lemma Phi_quotient_abelem : p.-abelem (P / 'Phi(P)).
Lemma Phi_Mho : abelian P -> 'Phi(P) = 'Mho^1(P).
Lemma morphim_Phi rT P D (f : {morphism D >-> rT}) :
p.-group P -> P \subset D -> f @* 'Phi(P) = 'Phi(f @* P).
Lemma quotient_Phi P H :
p.-group P -> P \subset 'N(H) -> 'Phi(P) / H = 'Phi(P / H).
Lemma Phi_cprod G H K :
p.-group G -> H \* K = G -> 'Phi(H) \* 'Phi(K) = 'Phi(G).
Lemma Phi_mulg H K :
p.-group H -> p.-group K -> K \subset 'C(H) ->
'Phi(H * K) = 'Phi(H) * 'Phi(K).
Lemma charsimpleP G :
reflect (G :!=: 1 /\ forall K, K :!=: 1 -> K \char G -> K :=: G)
(charsimple G).