(* The two other assertions of Aschbacher 23.15 state properties of the       *)
(* normal series 1 <| Z = 'Ohm_1(A) <| A with A \in 'SCN(G).                  *)
Section SCNseries.
Variables A : {group gT}.
Hypothesis SCN_A : A \in 'SCN(G).
Let Z := 'Ohm_1(A).
Let cAA := SCN_abelian SCN_A.
Let sZA: Z \subset A := Ohm_sub 1 A.
Let nZA : A \subset 'N(Z) := sub_abelian_norm cAA sZA.

(* This is Aschbacher 23.15(3); note that this proof does not depend on the   *)
(* maximality of A.                                                           *)
Lemma Ohm1_stab_Ohm1_SCN_series :
odd p -> p.-group G -> 'Ohm_1('C_G(Z)) \subset 'C_G(A / Z | 'Q).