End HallCorollaries.
Section InternalAction.
Variables (pi : nat_pred) (gT : finGroupType).
Implicit Types G H K A X : {group gT}.

(* This is B & G, Proposition 1.5(a) *)
Proposition coprime_Hall_exists A G :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
exists2 H : {group gT}, pi.-Hall(G) H & A \subset 'N(H).