End Gaschutz.

(* This is the TI part of B & G, Proposition 1.6(d).                          *)
(* We go with B & G rather than Aschbacher and will derive 1.6(e) from (d),   *)
(* rather than the converse, because the derivation of 24.6 from 24.3 in      *)
(* Aschbacher requires a separate reduction to p-groups to yield 1.6(d),      *)
(* making it altogether longer than the direct Gaschutz-style proof.          *)
(* This Lemma is used in maximal.v for the proof of Aschbacher 24.7.          *)
Lemma coprime_abel_cent_TI (gT : finGroupType) (A G : {group gT}) :
A \subset 'N(G) -> coprime #|G| #|A| -> abelian G -> 'C_[~: G, A](A) = 1.