1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

874 13. MID-SIZE GROUPS OVER F2

ce~tralizes OP(F(Gu)) and either X = Rp::::; Op( Gu), or X centralizes Op( Gu) and

hence F(Gu)·

Let z E z#. By 1.1.6, the hypotheses of 1.1.5 are satisfied with (u), Gu, Tu,

Mc in the roles of "B, H, 8, M". By the claim, O(F(Gu))::::; Na,,.(X)::::; Mu,

so O(F(Gu))::::; Ca,,.(z) for z E Zn 02 (X)#. But by 1.1.5.2, z inverts O(Gu), so

O(Gu) = 1.

Assume that 02 ,F* (Gu) ::::; NaJX). Then X centralizes E(Gu)· We saw that

X centralizes OP(F(Gu)) and either X = Rp or X centralizes F(Gu)· In the

latter case X centralizes F*(Gu), so that X::::; Op( Gu), and then as mp(X) = 2 =

mp(D1(0p(Gu))), X = D1(0p(Gu)). Hence in either case Gu ::::; Na,,. (X) =Mc,

contrary to assumption.

· Therefore there exists KE C(Gu) with Ki Na(X) and K/02(K) quasisimple.

Then X = 02 (X) normalizes K by 1.2.1.3, and K = [K, X] by A.3.3.7. Set

Ko:= (KTu).

Suppose NM,,. (K) is irreducible on X/0 2 ,q;,(X). Then Ox(K) ::::; 02,q;,(X) and

as K is described in Theorem C (A.2.3), mp(Out(K)) ::::; 1 since p > 3. Therefore

since NMJK) is irreducible on X/0 2 ,q;,(X), X induces inner automorphisms on k.

Then mp(K) > 1, so K =OP' (Gu) by A.3.18. Thus Ko =Kand X::::; K. Also

TuX = XTu with Tu E 8yb(Gu), so Kand the embedding of X in Kare described

in A.3.15. As mp(Autx(K)) > 1, and NM,,.(K) is irreducible on X/02,<P(X),

conclusion (3) of A.3.15 is eliminated, so conclusion (2) or (5) of A.3.15 holds.

Let PE Sylp(X). During the proof of (1) and (2), we saw that Tu is reducible

on X/0 2 ,q;,(X) in case (2) of 13.1.10, and in the remaining cases Tu is irreducible

and AulT,,.nL(P) is noncyclic. Suppose Tu is irreducible on X/02,<P(X). Then

XE 3(Gu, Tu)· We observe that the proof of 1.3.4 does not require the hypotheses

that H E 'H(XT), but only that H E 'H(X), and NTnH(X) is irreducible on

X/02,q;,(X). Thus we may apply the analogue of that result with Gu, Tu, Kin the

roles of "H, T, (LT)", to conclude that k is described in 1.3.4. Therefore if A.3.15.5

holds, then K ~ 8p4(2n) with AutT,,. (P) cyclic, contary to a remark earlier in the

paragraph. Thus Tu is reducible on X/0 2 ,q;,(X), so we are in case (2) of 13.1.10,

where CL(u)+ contains an 83 -section. In case (2) of A.3.15, Tu is irreducible on

X/02,q;,(X), so we are in case (5) of A.3.15. Then as P::::; K with PTu = TuP and

p > 3, k is of Lie type over F 2 n with 2n = 1 mod p, and P is contained in the Borel
subgroup NK(Tu n K). Hence the 83-section is induced by outer automorphisms
of k, so from the structure of Out(K 0 /0 2 (K 0 )), k ~ (8)L 3 (2n) with n even.
Having discussed the case where NM,,. (K) is irreducible on X/0 2 ,q;,(X), we now

turn to the remaining case where it is reducible. If Tu normalizes K, then so does

Mu by 1.2.1.3, and then (1) contradicts the assumption in this case. Therefore

K <Ko. However Mu acts on Ko and is irreducible on X/0 2 ,q;,(X) by (1). Further

by 1.2.1.3, Out(K) is cyclic. so as Out(Ko) is Out(K) wr Z 2 , again X induces

inner automorphisms on K 0. By 1.2.2.a, Ko = OP' (Gu), so a Sylow p-subgroup

P of X is containied in Ko and P = PK x Pf<, where PK := P n K ~ Zp and

t E Tu - NTu (K). As TuP = PTu, we conclude from 1.2.1.3 and A.3.15 that K is

isomorphic to L 2 (2n) or 8z(2n) with 2n = 1 mod p, J 1 with p = 7, or L 2 (r) for a

suitable odd prime r.

Summarizing our list of possibilities, X ::::; Ko and either