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(d): the more traditional form of the lemma *)
(* above, with the assumption H <| G weakened to H \subset G. The stronger   *)
(* coprime and solvability assumptions are easier to satisfy in practice.    *)
Proposition coprime_quotient_cent A G H :
H \subset G -> A \subset 'N(H) -> coprime #|G| #|A| -> solvable G ->
'C_G(A) / H = 'C_(G / H)(A / H).