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

(* A weaker but more practical version, still stronger than the usual form *)
(* (viz. Aschbacher 18.7.4), similar to the one needed in Aschbacher's     *)
(* proof of Thompson factorization. Note that the coprime and solvability  *)
(* assumptions could be further weakened to H :&: G (and hence become      *)
(* trivial if H and G are TI). However, the assumption that A act on G is  *)
(* needed in this case.                                                    *)
Lemma coprime_norm_quotient_cent A G H :
A \subset 'N(G) -> A \subset 'N(H) -> coprime #|H| #|A| -> solvable H ->
'C_G(A) / H = 'C_(G / H)(A / H).