End ExternalAction.
Section SylowSolvableAct.
Variables (gT : finGroupType) (p : nat).
Implicit Types A B G X : {group gT}.

(* Specifications: *)
Lemma sol_coprime_Sylow_exists A G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
exists2 P : {group gT}, p.-Sylow(G) P & A \subset 'N(P).
Lemma sol_coprime_Sylow_trans A G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
[transitive 'C_G(A), on [set P in 'Syl_p(G) | A \subset 'N(P)] | 'JG].
Lemma sol_coprime_Sylow_subset A G X :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable A ->
X \subset G -> p.-group X -> A \subset 'N(X) ->
exists P : {group gT}, [/\ p.-Sylow(G) P, A \subset 'N(P) & X \subset P].