1006 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN Cf(G, T) IS EMPTYTheorem 14.3.26 handled the case where U is nonabelian but not extraspecial,
so this section will complete the treatment of the case U nonabelian. Recall also
by Theorem 14.3.16 that L/0 2 (£) ~ £3(2).Recall that in Hypothesis 14.3.10, H := G 1 , U = (V^01 ), and we can appeal to
results in both subsections of section 12.8. Also g E N L(Vi) - H and W := Cu(Vi).
Let s be the generator of Vf. As U is extraspecial, Zu = Vi, so that fj = U,
Zf; = 1, iI = H*, and d := m(U) = m(U). By 12.8.4.4, Q := 02(H) = CH(U).Let K := 02 (H). By 12.8.8.2, H preserves a quadratic form on U, so H :S;
O(U) ~ Od(2) for E := ±1. Notice CHcV2) = NH(Vi) :S; G2, so since I2 :SI G2 by
12.8.9.1, CH* (V2) acts on W9* by 12.8.12.2, and on jj; since E = WnW^9 = WnW^1 '
where V{ is the point of Vi distinct from V1 and Vf.
As Zf;* = 1, 12.8.11.5 becomes:LEMMA 14.4.1. m(E) + m(WB*) ~ m(U) - 1 = d - 1.
We next obtain a list of possiblities for H* and U from G.11.2; all but the
second case will eventually be eliminated, although several correspond to shadows
which are not quasithin.LEMMA 14.4.2. m(E) - = d/2, so m(WB*) = d/2 - 1, U ~ Qd~
8 , and one of the
following holds:
{1) d = 4 and H* ~ 83 x 83.
(2) d = 4 and H* is E 9 extended by Z 2 •
{3) d = 8, U is the natural module for K* ~ nt(4), and W^9 * = Cr•nK• (x*)(;r*),where x E W^9 - K interchanges the two components of K, and m([U, x*]) = 4.
(4) d = 8, H* ~ 87' u = U1 EB U2, where ui is a totally singular K -module of
rank 4, and U'f = U2 for x E WB - NH(U1).(5) d = 8, H* ~ 83 x 85 or 83 x A5, and U is the tensor product of the natural
module for 83 and the natural or A5-module for L2(4).
(6) d = 12 and H* ~ Z2/M22·PROOF. Notice the assertion that m(WB*) = d/2 - 1 will follow from 14.4.1
once we show m(E) = d/2, as will the assertion that U ~ Q:/
2
.
By 14.3.13, H* and its action on U satisfy one of the conclusions of G.11.2,
but not conclusion (1), (4), (5), or (12). Further by 14.3.23: d :'.::: 4, and if d = 4
then E = V is of rank 2 = d/2, so that either (1) or (2) of 14.4.2 holds, since inconclusions (ii) and (iii) of 14.3.23.2, 1 =/= Zf;*, contrary to an earlier remark.
Suppose d = 6. Then conclusion (3) or (6) of G.11.2 holds. In either case,
27 divides the order of H*, so E = -1 as 27 does not divide the order of Ot(2).
Therefore m(E) :S; m 2 (U) = 3, so E = V is of rank 2 as V :S; E by 12.8.13.1,
·and hence m(WB*) = 3 by 14.4.1. Thus conclusion (3) of G.11.2 does not hold, as
there m2(H*) = 2. In conclusion (6), CH•(V 2 ) acts on E of rank 2, impossible as
CH* (V2) is the stabilizer in H* of a point of U, and so acts on no line of U.In the remaining cases of G.11.2, we have d = 8 or 12. So m(WB*) = d/2 - 1
by 14.3.14, and thus m(E) = d/2 by 14.4.1, completing the proof of the initial
conclusions of the lemma as mentioned earlier.
If d = 12, then conclusion (13) of G.11.2 holds, and hence conclusion (6) of
14.4.2 holds. Thus we may assume one of conclusions (7)-(11) of G.11.2 holds,
where d = 8.