14.3. FIRST STEPS; REDUCING (V^0 1) NONABELIAN TO EXTRASPECIAL 999Let G1 := GA/A. From the structure of Aut(K) in 14.3.19.10, since A =
Ca( A), G1~85 with d+ = (5, 6). Recall g E NL(V2)-H, so that w := dd9d9
2
is an
involution with w+ = (1, 2)(3, 4)(5, 6), and hence X ~ Z1xZ 2 , with D 1 (XnU) = V.
Now I2 acts on D1(X) = V x (w/, so as [A,w] = V and A::::; R, [R,w] = V.
Therefore R is transitive on V w, so by a Frattini Argument, J 2 = RCh ( w), and
hence Cr2(w)/CR(w) ~ 83. Further ICR(w)I = IRl/4 = IX n RI, so CR(w) =
X n R ~ Zl Also for u EU - R, d+gd+g2
= [d+9,u], so d9d9
2
= [d9,u] mod V
since V = XnA. Then [d^9 , u] E (XnU)-V, so [dg, u] is oforder 4 as !1 1 (XnU) = V.
Thus d9d92
EU has order 4 and hence as 02 (H) centralizes d,
2
Co2(H)(w) = Co2(H)(d^9 d^9 ) ~ Z4 * SL2(3).
Further choosing Tso that Tw := CT(w) E Syl 2 (CH(w)), D 1 (Z(Tw)) = (w, z/ and
WZ E WU.
Set Gw := Ca(w). As CR(w) ~ Z1, 02 (Cr 2 (w)) ~ Z3/Z1, while by (5) of
14.3.18, 02(0^2 (H)) = U ~ Q~ has no Z1-subgroup, so we conclude w ¢'. z^0.
Thus as D 1 (Z(Tw)) = (w, z/ and wz E w^0 , z is weakly closed in Z(Tw), so that
Na(Tw)::::; Hand hence Tw E Syl2(Gw)·
If z E 02(Gw), then V = (z^0 I2(w)l ::::; Z(02(Gw)), impossible since V i.Z((VOH(w)/). Thus z i;i 02(Gw); now Tw E Syb(Gw), 02(CH(w)) ::::; GA, and
z is contained in each nontrivial normal subgroup of Gw nGA other than (w/, so we
conclude that 02(G 2 ) = (w/. As in the the proof of 14.3.20, we appeal to 1.2.11,
1.1.6, and 1.1.5.3; this time from the structure of CH(w) = Ccw(z) and Cr 2 (w), we
conclude Gw/ (w/ ~ G2(2), so Gw ~ Z 2 x G 2 (2) since G2(2) has trivial Schur multi-plier by I.1.3. Set Lw := G':, and observe that Lw has one class zLw of involutions,
and so the set { w }u(zw)Lw of involutions in wLw is contained in w^0 since we saw wis conjugate to zw. Also Tw n (w/Lw = XCu(t) ::::; RQ, so Tw n (w/Lw = Tw n RQ.
By 14.3.21.2, no involution in T - RQ is in z^0 , so z^0 n Gw = z^0 w, and hence
w^0 n H = wH since G is transitive on commuting pairs from z^0 x w^0. But then as
H/0^2 (H)R is of order 4 and w ¢'.RU, it follows that w ¢'. 02 (G) from Generalized
Thompson Transfer A.1.37.2, contrary to the simplicity of G. D
We are now in a position to derive a contradiction, and hence establish Theorem
14.3.16. By 14.3.22, Q = U, so IT : RUI = 2. Thus by 14.3.21.1, there is an
involution t E T - RU. By 14.3.21.2, t ¢'. z^0 , while by 14.3.21.3, all involutions in
RU are in z^0. Thus t^0 n RU= 0, sot¢. 02 (G) by Thompson Transfer, contrary
to the simplicity of G.
14.3.4. Characterizing HS by (VG^11 nonabelian but not extraspecial.
In this subsection we continue to assume Hypothesis 14.3.10. By Theorem 14.3.16,
case (1) of Hypothesis 14.3.1 holds. Thus in the remainder of our treatment of the
case U nonabelian in this section and the next, we have L/02(L) ~ L3(2).
In this final subsection, we first prove several more preliminary results, and
then reduce to the case where U is extraspecial, by showing HS is the only qua-
sithin example with V 1 < Zu. The treatment of the extraspecial case occupies the
following section 14.4.
LEMMA 14.3.23. d ~ 4. If d = 4, then
(1) v = E ~ E4.