14.3. FIRST STEPS; REDUCING (VG1) NONABELIAN TO EXTRASPECIAL 993rank 1. Thus for either choice of i = 1, 2, there exists 9i E L with B9i :::; T but
B9i f:. Ri· Hence (3) holds. 0
14.3.2. Preliminary results for the case (VG^1 ) is nonabelian. When
(V^01 ) is nonabelian, we will concentrate on G 1 , as opposed to an arbitrary memberof Hz; recall the latter set was defined in Notation 12.8.2.3. Thus in the remainder
of this section, and indeed in the subsequent section 14.4, we assume:HYPOTHESIS 14.3.10. Assume Hypothesis 14.3.1 with U := (V^01 ) nonabelian.
Take H := G1.
Observe that U plays the role of "UH" in Notation 12.8.2; in particular by
12.8.4.2, U is elementary abelian.
Since U is nonabelian, we also adopt the notation of the second subsection
of section 12.8. Since H f:. M, V < U. Write Q := 02 (H), rather than QH as
in section 12.8, set H* := H/Q, Zu := Z(U), fI := H/Zu, iI := H/CH(U), pick
g E NL(Vi)-H, let I2 := (UL^2 ), W := Cu(Vi), and E := WnWg. Let d := m(U).
By 12.8.8.1, U = UoZu with Uo extraspecial and cI>(Uo) =Vi, and iI preserves
a symplectic form on U of dimension d. By 12.8.12, this action satisfies Hypothesis
G.10.1, with if, u, ~, E, W^9 , zg in the roles of "G, v, Vi, w, x, Xo", and
Hypothesis G.11.l is also satisfied. Thus we may make use of results from sections
G.10 and G.11. Recall also from G.10.2 that the bound(*) of sections G.7 and G.9
holds, so that we may apply the results of section G.9.
By 12.8.8.3:
LEMMA 14.3.11. m(V) = m(V).LEMMA 14.3.12. Assume m(Wg) :::; d/2 - 1. Then
(1) m(Wg) = d/2 - i.
(2) m(E) = d/2.
(3) Zu = V 1 , so U is extraspecial, U = U, and iI = H*.
(4) H preserves a quadratic form on U of maximal Witt index in which E is
totally singular.
PROOF. As m(W9) :::; d/2-1, the first inequality in G.10.2 is an equality with
Ziji = 1. Thus (1) holds. Then (1) and 12.8.11.5 imply (2). As Ziji = 1, Zu =Vi by
12.8.13.4. Thus U is extraspecial by 12.8.8.1, so U = U. By 12.8.4.4, Q = CH(U),
so H* =if. Thus (3) holds. Also cI>(Zu) = cI>(V 1 ) = 1, so by 12.8.8.2, H preserves a
quadratic form q(ii) := u^2 on U. By 12.8.11.2, cI>(E) = 1, so Eis a totally singular
subspace of the orthogonal space U, ofrank d/2 by (2). Thus U is of maximal Witt
index, completing the proof of (4). 0
LEMMA 14.3.13. iI and its action on U satisfy one of the conclusions of The-
orem G.11.2, but not conclusion (1), (4), (5), or {12).
PROOF. By 12.8.12.4, iI and its action on U satisfy one of the conclusions
of G.11.2. By (6) and (7) of 12.8.13, conclusions (4) and (12) are not satis-
fied. If conclusion (5) is satisfied, then by 12.8.13.5, there is K E C1(G, T) with
K/02(K) ~As, contrary to 14.3.4.1.
Assume conclusion (1) is satisfied. Then d = 2 and if ~ 83. By 14.3.11,
m(V) = m(V), so if L/0 2 (£) ~ £ 3 (2), then m(U) = 2 = m(V), and hence