End ModularGroupAction.
Section Sylow.
Variables (p : nat) (gT : finGroupType) (G : {group gT}).
Implicit Types P Q H K : {group gT}.

(* Specifications: *)
Theorem Sylow's_theorem :
[/\ forall P, [max P | p.-subgroup(G) P] = p.-Sylow(G) P,
[transitive G, on 'Syl_p(G) | 'JG],
forall P, p.-Sylow(G) P -> #|'Syl_p(G)| = #|G : 'N_G(P)|
&  prime p -> #|'Syl_p(G)| %% p = 1%N].
Lemma max_pgroup_Sylow P : [max P | p.-subgroup(G) P] = p.-Sylow(G) P.
Lemma Sylow_exists : {P : {group gT} | p.-Sylow(G) P}.
Lemma Sylow_subJ P Q :
p.-Sylow(G) P -> Q \subset G -> p.-group Q ->
exists2 x, x \in G & Q \subset P :^ x.
Lemma Sylow_Jsub P Q :
p.-Sylow(G) P -> Q \subset G -> p.-group Q ->
exists2 x, x \in G & Q :^ x \subset P.
Lemma card_Syl P : p.-Sylow(G) P -> #|'Syl_p(G)| = #|G : 'N_G(P)|.
End Sylow.