14.3. FIRST STEPS; REDUCING (VG1) NONABELIAN TO EXTRASPECIAL 995Next by 12.8.13.3, z& 2'! Zu, so z& -=I-1. Let K := 02 ((z&H)). By 12.8.13.3,
z& centralizes Zu, so K centralizes Zu. Thus using Coprime Action, 02 (CK(U))
centralizes U, and hence 1 -=/:-k 2'! K* by parts (2) and (4) of 12.8.4. So if iI 2'!
83 x 83 then K :S Mv by 14.3.5 and 14.3.3.6, so K centralizes V of order 2. But
then as K :::l H, K centralizes U = (VH), contrary to k-=/:-1.. +.....
Therefore H 2'! 04 (2). By 12.8.12.2, z& :::l T, so 03 (H) = [0 3 (H), Z&J,
and hence 03(H) :S k. Thus K* 2'! E9, so U = [U, K]. Then ~s K centralizes
Zu, [J = [U,K] EB Zu with Zu -=I- 0. Further as z& :::l T and w::; Ca(V) :S
Ca(Zf), [Z&, W] :S Z& n W. As L/02(L) 2'! L2(2)^1 , Z(I 2 ) = 1by12.8.13.3. Thus
z& n W = Vl by 12.8.10.3, so as z& acts nontrivially on the hyperplane W of (jand centralizes Zu,
v = Vl::; [W, Z&J::; [U, K],
so U = (VH)::; [U,K], contradicting 0-=/:-Zu f:. [U,K].
is complete.Thus the proof of 14.3.15
D14.3.3. Eliminating L 2 (2) when (VG^1 ) is nonabelian. Recall that (V^01 )
is nonabelian in the quasithin examples for L/0 2 (L) 2'! L2(2)' characterized in
section 14.2; but of course those groups are now excluded in Hypothesis 14.3.1.
Thus in this subsection we prove:THEOREM 14.3.16. Assume Hypothesis 14.3.10. Then case (1) of Hypothesis
14.3.1 holds, namely L/0 2 (L) 2'! L 3 (2).
REMARK 14.3.17. In proving Theorem 14.3.16, we will be dealing in effect only
with the shadows of extensions of U 4 (3) which interchange the two classes of 2-
locals isomorphic to A 6 / E 16. These extensions satisfy our hypotheses except they
are not simple, and sometimes not quasithin. Thus we construct 2-local subgroups
which appear in those shadows, and eventually achieve a contradiction by showing
02 ( G) < G using transfer.
Until the proof of Theorem 14.3.16 is complete, assume G is a counterexample.
Thus case (2) of 14.3.1 holds, so V1 = Z = (:z), V = V2 and G2 = Na(V) = M.
Recall Q = 02 (H). By 14.3.15, U 2'! Q~, and Uhas an orthogonal structure over
F 2 preserved by H = G1, with H* = H/Q = O(U) 2'! ot(2) and E totally singular.
Thus H is a {2, 3}-group, so in particular, H is solvable.
Recall I 2 = (UL^2 ) = (UL), and by 12.8.9.1, I 2 :::l G2 =Mand L = 02 (! 2 ).
LEMMA 14.3.18. (1) V = Q n UB.
(2) I2 :::l M, R := 02(h) = 02(L).(3) R is the 4-subgroup of T containing no transvections, and hence lying in
nt(u); so IT: RQI = 2.
(4) R = AAt, where A 2'! E 16 and At are the maximal elementary abelian
subgroups of R, IRI =2^6 , tET-RQ, V=AnAt, andA :::l hQ.
(5) U = 02(0^2 (H)).
(6) NH(A) =CH (A) 2'! Z2 x 83.
PROOF. First h plays the role of "I" in 12.8.8; then by 12.8.8.4 we may apply
G.2.3.4 to conclude that E = W n WB is T-invariant. But we saw E is totally
singular and H = O(U) 2'! ot(2), so Tacts on no totally singular 2-subspace of