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

(* This is B & G, Proposition 1.5(b). *)
Proposition coprime_Hall_subset pi (gT : finGroupType) (A G X : {group gT}) :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
X \subset G -> pi.-group X -> A \subset 'N(X) ->
exists H : {group gT}, [/\ pi.-Hall(G) H, A \subset 'N(H) & X \subset H].