(^864) 12. LARGER GROUPS OVER F2 IN Cj(G, T)
for u E UH, m(B/CB(u)) S m(V 1 ) = 1, so CB(u) is noncyclic, and hence by 12.9.7,
u E Na(A). Thus UH S Na(A), and so [UH, B] SA n V1=1.
As 02 (H) S (AH) and Vi <UH, there is h EH such that Ah does not act on
Vi. But again using 12.9.7,
D := Bh S Ca(UH) S Ca(V2) S Na(V).
If [D, V] = 1, then Vs Ca(D) s Na(Ah) by 12.9.7, so Ah s Ca(V) S Ca(Vi)
by 12.9.9.1, contrary to our choice of h. Thus fJ -=!= 1, so D = Vgh n M by 12.9.9.2
However
1-=!= [UH, Ah] s UH nAh s CD(V),
since UH is abelian. As D is a hyperplane of Ah with D = vgh n M, 12.9.9.2
supplies a contradiction.
This final contradiction completes the proof of Theorem 12.9.1.