14.6. ELIMINATING L 2 (2) WHEN (Val) IS ABELIAN 1031normal subgroup of H*. Thus Tis maximal in PT= H, and P ~ Zp or EP2, since
H is an SQTK-group.
Set K := 02 (H), so that K = P.
LEMMA 14.6.12. (1) p = 3 or 5.
( 2) There is a subgroup Ho of index 2 in H such that H 0 = HJ. x H2, Ht ~ D2p,
H2 =Hf fort ET - NT(H1) and Hi the preimage of Ht in H, and [UH,H] =U H,1 EB U H,2, where U H,i := [UH, Hi] is of rank 4 when p = 5, and of rank 2 or 4
whenp = 3.
(3) ZS [UH,0^2 (Hi)] for each i.
PROOF. By 14.5.18.3, q(H*,UH) s 2. Let Ho := (Q*(H*,UH)); as Tis
maximal in H, H = HoT. By D.2.17, H 0 = HJ. x · · · x H; and [UH, Ho] =UH,1 EB··· EB UH,s, where (Ht, UH,i) are indecomposables in the sense of D.2.17. In
particular p = 3 or 5 by D.2.17, so that (1) holds. Further Op(H 0 )* is not of order
p by 14.3.5. Hence P* ~ Ep2, and as T is irreducible on P*, our indecomposablesappear only in conclusions (1) or (2) of D.2.17, so that (2) holds. Finally (3) follows
from 14.6.2. DDuring the remainder of the proof of Theorem 14.6.11, we adopt the notationof 14.6.12.2, with UH,i the preimage in UH of UH,i· Also set UK:= [UH, H].
LEMMA 14.6.13. Either
(1) p = 3, UK is a 4-dimensional orthogonal space over F 2 forH* = O(UK) ~ ot(2),
and [UH,1,UJ!r] =/= 1 for some g E G-G1 such that UJ!r S NH(UH,1) and UH S H^9 ,
or
(2) p = 3 or 5, m(UK) = 8, D-y < U'Y, and we may choose 'Y so that u; S HJ.,
Z'Y S UH,1, and ZS UJ!r , v for g E G - G1 with "(1g = 'Y·PROOF. Suppose first that D'Y = Uy. Then by 14.5.18.1, UH induces a non-
trivial group of transvections on U'Y with center Z, so by 14.6.12, p = 3, and H*
acts as ot(2) on UK of rank 4. Since b 2: 3 is odd by F.9.11.1, in this case there
is g E (LT, H) with 'Yl = 7g. Then UJ!r induces a nontrivial group of transvectionson UH with center Z^9 , so UJ!r S NH(UH, 1 ), and we may choose notation so that
[UH,1, UJ!r] =/= 1. By F.9.13.2, U'Y SH, so UH = U~ S H^9. Thus (1) holds when
D'Y = U'Y.Hence we may suppose instead that D'Y < U'Y. So by 14.5.18.4, we may choose 'Y
with m(U;) 2: m(UH/DH) > 0 and u; E Q(H*, UH); in particular u'Y is quadratic
on UH, and hence either u'Y acts on UH,1, or else the quadratic action forces u; =(x*) to be of order 2 with UJ},i = U H, 2. Let g E (LT, H) with 71 g = 'Y·
Suppose first that u; = (x*) is of order 2 with UJf, 1 = UH,2· As u; E
Q(H*, UH),
m(UH/CuH(U'Y)) s 2m(U;) = 2,while CuH(U'Y) = [UH,U'Y] since x* is an involution with UJf, 1 = UH,2· Therefore
m(UH) = 4, and the inequality is an equality. Again by 14.6.12, p = 3, UK is a
4-dimensional orthogonal space over F 2 , and H* = O(UK)· Further Z'Y = [U'Y,DH]
by F.9.13.6, so z9 is a singular vector in UK since ujl; 1uu'f;2 is the set of nonsingular
' '