End SolvablePrimeFactor.
Section Special.
Variables (gT : finGroupType) (p : nat) (A G : {group gT}).

(* Specifications: *)
(* This is Aschbacher (23.7) *)
Lemma center_special_abelem : p.-group G -> special G -> p.-abelem 'Z(G).
Lemma exponent_special : p.-group G -> special G -> exponent G %| p ^ 2.
(* Aschbacher 24.7 (replaces Gorenstein 5.3.7) *)
Theorem abelian_charsimple_special :
p.-group G -> coprime #|G| #|A| -> [~: G, A] = G ->
\bigcup_(H : {group gT} | (H \char G) && abelian H) H \subset 'C(A) ->
special G /\ 'C_G(A) = 'Z(G).