Section StructureCorollaries.
Variable S : {group gT}.
Hypotheses (pS : p.-group S) (esS : extraspecial S).
Let p_pr := extraspecial_prime pS esS.
Let oZ := card_center_extraspecial pS esS.

(* Specifications: *)
(* This is Aschbacher (23.10)(2). *)
Lemma card_extraspecial : {n | n > 0 & #|S| = (p ^ n.*2.+1)%N}.
Lemma Aut_extraspecial_full : Aut_in (Aut S) 'Z(S) \isog Aut 'Z(S).
(* These are the parts of Aschbacher (23.12) and exercise 8.5 that are later  *)
(* used in Aschbacher (34.9), which itself replaces the informal discussion   *)
(* quoted from Gorenstein in the proof of B & G, Theorem 2.5.                 *)
Lemma center_aut_extraspecial k : coprime k p ->
exists2 f, f \in Aut S & forall z, z \in 'Z(S) -> f z = (z ^+ k)%g.