1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

12.8. GENERAL TECHNIQUES FOR Ln(2) ON THE NATURAL MODULE 851

F.8.1 is satisfied for each H E Hz, while the remaining conditions are easily ver-

ified; for example, 12.8.4.4 says ker 0 HCV)(H) = QH, giving (c). Thus (3) implies

(4). Finally (4) implies (1) by F.8.5.2, and (4) and (5) are equivalent by Remark

F.9.9. D

REMARK 12.8.7. Notice that if one of the equivalent conditions in 12.8.6 holds,

then from condition (5), (VH) = (Vf). Thus the subgroups denoted by "UH" and

"VH" in section F.9 both coincide with the group denoted by UH in this section.

When UH is abelian for all H E Hz, by parts (4) and (5) of 12.8.6, we can

apply lemmas from sections F.8 and F.9 to analyze the amalgam defined by LT

and H. Notice in particular by F.8.5 and F.9:11 that in this case the amalgam

parameter "b" of those sections is odd and at least 3. On the other hand when UH

is nonabelian, we normally specialize to the case H = G 1 , and apply methods from

the theory of large extraspecial 2-subgroups, which are developed further in the

following subsection.

12.8.2. (VG^1 ) nonabelian almost extraspecial subgroups. In this sub-

section, we consider the case where (VG^1 ) is nonabelian. The analysis in the sub-

section continues to develop the theory of almost extraspecial 2-subgroups U (i.e.,

U is nonabelian and l<I>(U)I = 2) begun in section G.2 of Volume I. The theory is

a variant of the theory of large extraspecial 2-subgroups appearing in the original

classification literature.

In the remainder of the section we take H := G1 and assume that U :=UH=

(VH) is nonabelian.

As U is nonabelian, 12.8.4.3 says that: ,

(U) =Vi.

Set fl:= H/Z(U) and iI := H/CH(U).

LEMMA 12.8.8. (1) U = U 0 Z(U), with Uo an extraspecial 2-group and <I>(Uo) =

defines a symmetric bilinear form on fj with radical Z(ii) preserved by H*, which

induces an H-invariant symplectic form on U. If <I>(Z(U)) = 1, then