End ExponentPextraspecialTheory.
Section GeneralExponentPextraspecialTheory.
Variable p : nat.

(* This is the uniqueness half of Aschbacher (23.13); the proof incorporates *)
(* in part the proof that symplectic spaces are hyperbolic (19.16).          *)
Lemma isog_pX1p2n n (gT : finGroupType) (G : {group gT}) :
prime p -> extraspecial G -> #|G| = (p ^ n.*2.+1)%N -> exponent G %| p ->
G \isog p^{1+2*n}.