1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

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12.2. GROUPS OVER F2, AND THE CASE VA TI-SET ING 801


Suppose that An02(H) = 1. Then A~ A, so m(A) = m(V) 23by12.2.2.3.

But if H is solvable, then m2(H*) :::; 2 as H = 02 ,p, 2 (H) for some odd prime p

by B.6.8.2, so that H/02,p(H) is a subgroup of GL 2 (p). On the other hand if H

is nonsolvable, then by 12.2.7.2, K ~ L 2 (4), so that again m 2 (H) :::; 2. Thus in

either case, we have a contradiction to m(A*) 2 3.
This contradiction shows that 1 #-An 02 (H). Thus for each h E H, 1 #

Ahn 02(H):::; NAh(V) by 12.2.7.1. However as V:::; 02 (H), (V,Ah) is a 2-group,

so [V, Ah] = 1 by I.6.2.1. We saw K:::; (AH), so K:::; Ca(V) :::; M, contradicting

H 1:. M. This completes the proof of 12.2.18. D

LEMMA 12.2.19. H is solvable.

PROOF. Assume that His not solvable. Then by 12.2.7, K/0 2 (K) ~ K* ~


L2(4). By 12.2.18, V i=-1. As v :::; 02(M), V :::; 02(M n H) = T n K* E

Syb(K). Thus either V = T n K ~ E 4 , or V :::; K is of order 2. Pick

h EK - M, and let U := V n 02 (H), I:= (V, Vh), and W1 := 02 (1). Then either


JVI = 4 and J = K ~ L2(4), or V is of order 2 and J* ~ D2m, m = 3 or

5. As m(V) 2 3 > m(V*), U #-1. Then as U:::; 02 (H) :::; NH(Vh) by 12.2.7.1,

Nv(Vh) #-1. It follows from (a) and (c) of I.6.2.2 that W 1 := U x Uh is a sum of


natural modules for I/W1 ~I; in particular if J ~ D 2 m, an element of order m

is fixed point free on W1. If V* is of order 2, pick x E H n M of order 3 and let.
K1 := (VX,I) and w := uxw1; if IV*I = 4 let K1 :=I and w := W1. Thus in
either case K* = Kj.
We claim that K1 acts on W, and Wis elementary abelian: Suppose first that

.IVI = 2. We saw that U #-1 normalizes vx, and as (V, V*x) is a 2-group by

our choice of x, (V, vx) is a 2-group. Therefore V centralizes vx by I.6.2.1. Now

by symmetry between I= (V, Vh) and (Vx, Vh), (Vx, Vh) acts on ux x Uh, so


K 1 = (V, vx, Vh) acts on W = uuxuh and Wis elementary abelian. On the other.

hand if JV* I = 4 then K1 = I acts on W = U x Uh, completing the proof of the

claim.

Next by 12.2.7.1, 02 (H) acts on V and Vh, and also on vx when JV*[ = 2,

so 02 (H) acts on K1 and W. Thus K02(H) = K102(H) acts on K1 and W.

Therefore as K 102 (H)/K1 ~ 02 (H)/(0 2 (H)nK1) is-a 2-group, K = 02 (H):::; K1.
Now if JV*I = 4, then K 1 =I, so W = 02 (1) is a sum of J).atural modules for

K ~ KJ/W. Suppose on the other hand that V is of order 2. We saw earlier that

V centralizes vx; hence W = uxw1 = Cw(V)W1, so that W = Cw(i) x W1 for i

of order min I, which is fixed point free on W1; and Cw(i) = Cw(I) :::; Cw(V) =
uux = Cw(Vx). Thus Cw(I) = Cw(K1). As [W, VJ= U and [W, vx] = ux with
vvx abelian, T* n K* = V*V*x is quadratic on W. Also i is fixed-point-free on

W 1 , so by G.1.5 and G.1.7, W/Cw(I) is a sum of natural modules for K*.

. Now ZnV #-1, so 1 #-Vz := ((ZnV)K) E R2(KT) by B.2.14. As Vis a TI-set
in G by 12.2.16 and Ki. M 2 Na(V), Cv(K) = 1. As Zn V:::; 02(H) n V = U :::;


W, Vz :::; W, so by the previous paragraph 1 #-Vz / Cvz ( K) is a sum of natural

modules for K*.

By 12.2.7.3, K :::; Y E Cj(G, T), with Y described in case (i) or (ii) of that

result. Let Vy:= ((ZnV)Y) and Y := Y/Cy(Vy); then Vz::::; Vy. As Vz/Cvz(K)
is a sum of natural L 2 (4)-modules, case (i) of 12.2.7.3 cannot arise, since there
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