1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

688 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL

L x CM ( L) centralizes z and hence lies in H, contrary to 02 ( H n M) = 1. Thus

85 ~ M = Lt, so that M = LT.

Assume first that CR(K) = Z is of order 2. Thus M ~ 85/E16, with H =

K(t) ~ (8 3 x Z 2 )/Q~. Then as CR(K) = Z, CH(P) ::::; P, and Z = Cp(X) for

X E 8yl 3 (H); thus G satisfies the Hypothesis on page 295 of C. Ku in [Ku97].

(Note that the term Zz there is unnecessary, and also that Z1 in H/Q should read

Z 2 ). We next verify that G is of type M 22 as defined on p. 295 of that paper-

namely we show there exists z i= zd E P with m(P n pd)= 2: Let DL be of order

3 in NL(T n £), and pick d E Dz. Then Zs = (z, zd) and P n pd= Zs~ E 4 , as

PnPd = 1 and Zs= PnPdnV from the structure of Land its action on V. Thus

G is of type M 22 , so we may apply the Main Theorem ·of that paper to conclude

that G ~ M22-

So now we assume that CR(K) ~ E 4 , and it remains to derive a contradiction.

Then M ~ S 5 /E32, with Q ~ E3 2. As Cr(L) = 1by6.1.6.1, Q does not split over

Vas an £-module. Thus Q = J(T).

Next all involutions in P are fused into V in K, and all involutions in V are

fused in£, as are all involutions in L - V. Thus all involutions in L are conjugate
in G, and are fused to some j E P - L. Next j induces a field automorphism on
L/V, so all involutions in jL are conjugate in£. Let T 0 := P(TnL) = (j)(TnL),
so that all involutions in T 0 are in zG. Let r E CR(K)-Z. Then r E Q-V, and as
Q = J(T), M = Na(Q) controls fusion in Q by Burnside's Fusion Lemma A.1.35.
Hence r ¢:. zG. Therefore rG n To = 0, so by Thompson Transfer, 02 (G) < G,
contradicting simplicity of G. This completes the proof of 6.2.13. D

By 6.2.13, we may assume during the remainder of the section that Zs < V n

U =:Vu; in Theorem 6.2.19, we will obtain a contradiction under this assumption.

Let Zu := Z(U).

As V* has order 2 by 6.2.10.4, m(Vu) ::::; m(V n QH) = 3, so as Zs < Vu:

LEMMA 6.2.14. Vu = v n QH is of rank 3.

LEMMA 6.2.15. (1) U = Zu * Uo is a central product, where U 0 is extraspecial

of width at least 2 and rank at least 3.

(2) For v EV - U there exists g EH with v*v*9 not a 2-element, and for each

such g, Jv*v*^9 1=3 and (V, V^9 ) ~ 83 /Q~ with Vu VJ= 02 ((V, V9))::::; U.

(3) Zu::::; Z(K) and K* is faithful on U/Zu.

PROOF. By 6.2.10.3, 02(H) = 1, so by the Baer-Suzuki Theorem A.1.2, there
is g EH with vv*^9 not a 2-element. Then Vi Na(V9), and so Vu ::::; Nv(V9) < V,
so by 6.2.14, Vu = Nv(V^9 ) is of index 2 in V. Similarly VJ = Nvg (V) is of index

2 in V^9 , so part (2) follows from 6.2.6. As Zu centralizes Vu, it centralizes V by

6.1.10.2, so Zu centralizes K = (VH). Thus CK•(U/Zu)::::; 02 (K)::::; 02 (H) = 1

using 6.2.10.3, so that K* is faithful on U/Zu, completing the proof of (3). As

(U) ::::; Z of order 2 by 6.2.10.1, and U is nonabelian by (2), (U) = Z. We