Section MoreSylow.
Variables (gT : finGroupType) (p : nat).
Implicit Types G H P : {group gT}.

(* Specifications: *)
Lemma Sylow_setI_normal G H P :
G <| H -> p.-Sylow(H) P -> p.-Sylow(G) (G :&: P).
Lemma normal_sylowP G :
reflect (exists2 P : {group gT}, p.-Sylow(G) P & P <| G)
(#|'Syl_p(G)| == 1%N).
Lemma trivg_center_pgroup P : p.-group P -> 'Z(P) = 1 -> P :=: 1.
Lemma p2group_abelian P : p.-group P -> logn p #|P| <= 2 -> abelian P.
Lemma card_p2group_abelian P : prime p -> #|P| = (p ^ 2)%N -> abelian P.
Lemma Sylow_transversal_gen (T : {set {group gT}}) G :
(forall P, P \in T -> P \subset G) ->
(forall p, p \in \pi(G) -> exists2 P, P \in T & p.-Sylow(G) P) ->
<< \bigcup_(P in T) P >> = G.
Lemma Sylow_gen G : <<\bigcup_(P : {group gT} | Sylow G P) P>> = G.
End MoreSylow.