iooo i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY
(i) if~ 83 x 83 , with Li :'SI if. Further if Ziji-=/:- 1 then Ziji = C.HCV) is
H... ...
of order 2, and setting K := (Ziji ), K ~ 83, H = KLiT, and Ki M.
(ii) iI ~ 85 , fj is the L2(4)-module, and Z2 ~ Ziji :::; E(H).
(iii) iI ~ A 6 or 85, and m(Ziji) = 1 or 2.
.. • 2
(iv) His E 9 extended by Z2, L 1 :'SI H, and U ~ Q 8.
(3) m(W9 /Ziji) = 1 and Ziji centralizes V.
(4) H > (H n M)CH(U).
PROOF. By 12.8.13.1, V :::; E. By 14.3.11, m(V) m(V) = 2. But by
12.8.11.2, Eis totally isotropic in the symplectic space V, so 2 = m(V) :::; m(E) :::;
d/2, and hence d ;::::: 4. Further if d = 4, these inequalities are equalities, so (1)
holds.
Assume d = 4. By 12.8.11.5 and (1), m(W9 / Ziji) = 1, while by 12.8.13.2, Ziji
centralizes V, and then by 12.8.11.3, Ziji is the kernel of the action of W^9 on V.
Thus (3) is established. By 14.3.3.6, H n M acts on V; so if (4) fails, then iI acts
on V, contrary to fj = (VH) and d = 4. Thus (4) holds.
Observe that if iI :::; ot(2), then 02 (H) is abelian, so Li :'SI 02 (H). Thus
Li :'SI 02 (H)T = iI. If 02 (H) is of order 3, then iI = LiT, contrary to (4). Thus
02 (H) ~ E 9 , so as Li :'SI if, we conclude iI < ot(2) in this case.
Suppose first that m 2 (H) = 1. Then by (3) and (4) of 14.3.12, U ~ Q§, so
iI :::; ot(2). Then by the previous paragraph, 02 (H) ~ E 9 , so as m 2 (H) = 1,
(2iv) holds.
Thus we may assume m 2 (H) ;::::: 2. Suppose first that iI :::; ot(2). Then
iI ~ 83 x 83 by remarks in paragraph three. Assume that Ziji -=/:-1. Then as Ziji
centralizes V by (3), and as V = [V,Li], Ziji is of order 2, Li = C 0 2(.H)(Ziji),
.. H.. ..
K := (Ziji ) ~ 83, and H = LiKT, and so K i M by (4). This completes the
proof that ( 2i) holds
Thus we may assume that iI is not contained in ot(2). But by 14.3.14.2, iI
is a subgroup of 8p4(2) containing 83 x 83 or A 5 , so we conclude F*(H) .~ L2(4)
or A5.
Suppose Zu =Vi. Then U is extraspecial, so iI:::; 0.4(2), and by the assump-
tion in previous paragraph, E = -1. This is impossible, as U contains the totally
singular line V. We conclude Zu >Vi, so Ziji-=/:-1by12.8.13.4.
Suppose F(H) ~ L2(4). As Li :'SI LiT and V = [V,Li] ~ E 4 , it follows that
fj is the £ 2 (4)-module, and Vis the F 4 -line invariant under T. As W9 is nontrivial
on V by 12.8.11.3, iI ~ 85. Then as W9 is elementary abelian and we saw that
1 -=/:-Zij < W9 and Zij centralizes V, (2ii) holds. A similar argument shows (2iii)
holds if F(H) ~ A 6. D
LEMMA 14.3.24. Assume Vi < Zu, so that U is not extraspecial. Then either:
(1) d = 6 and fj is the natural module for iI ~ G 2 (2), or
(2) d = 4 and one of conclusions (i)-(iii) of 14.3.23.2 holds.PROOF. By 14.3.12.3, m(W9) ;::::: d/2, so case (1) of 14.3.14 does not hold.
Case (3) of 14.3.14 is conclusion (1), and in case (2) of 14.3.14, d = 4 so one
of the conclusions of 14.3.23.2 holds, with conclusion (iv) ruled out as there U is
extraspecial. D