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(e). *)
Proposition coprime_comm_pcore A G K :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
pi^'.-Hall(G) K -> K \subset 'C_G(A) ->
[~: G, A] \subset 'O_pi(G).