1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

918 13. MID-SIZE GROUPS OVER F2

It remains to treat the case where [V3, VB] = V 1 = [V/, V]. Here m(VB / Cvg (V))
= 1, so V induces a transvection with center Vi on VB, and so again [V:f, V] = 1,
contrary to assumption. This contradiction completes the proof of 13.5.7. D

NOTATION 13.5.8. Recall Ch = GifVi_. By 13.5.7, we can appeal to the results

of section F.9 with V3 in the role of "V+" in F.9.1. Recall from Hypothesis F.9.1 that

for HE 1iz, UH:= (Vl), VH := (VH), and QH := 02(H). By F.9.2.1, UH:::; QH,

and by F.9.2.2, <!>(UH):::; Vi_. By F.9.2.3, QH = CH(UH); set H* := H/QH.

Notice G 1 n G 3 :::; M by 13.5.5, and H 1. M, so that:

LEMMA 13.5.9. Vi< UH

We begin our treatment of the case (V^01 ) nonabelian by considering the sub-

case where (V 3 °^1 ) is nonabelian; the next observation shows that if n = 5 and (V^01 )

is nonabelian, then (V3°^1 ) is also nonabelian:

LEMMA 13.5.10. If n = 5 and H E 1iz, then the following are equivalent:

(1) UH is abelian..

(2) VH is abelian.

(3) V:::; QH•

PROOF. When n = 5, CM(Vi) = CM(V), so the lemma follows from F.9.4.3.

D

LEMMA 13.5.11. If n = 5 and Vi :::; V n VB, then [V, VB] = 1 or Vi; in either

case, VB :::; Mv.

PROOF. We may assume [V, VB] =f. 1. By hypothesis V 1 :::; V n VB, so by

13.3.11.l we may take g E G1. By 13.3.11.5, [Vs, V/] =f. 1, so by F.9.2.2, [Vs, V:f] =

Vi_. Thus X := VsV:f ~ Dg x Z2, and A(X) ={Vi, V:f}. Now V:f acts on Vs, and

also V/ :::; Ua 1 :::; Mv, so [V, V/] :::; Vs :::; X, and hence V acts on X. Then as

A(X) = {Vs, Vi}, V :::; Na 1 (V:f) :::; Na(VB) by 13.2.3.2. By symmetry VB acts on

V, so [V, VB] :::; V n VB :::; Cv (VB). As [Vs, V:f] = Vi is singular, VB does not induce

a transvection on V, so m(Cv(VB)) :::; 2:::; m([V, VB]), and hence [V, VB] = V n VB

is of rank 2. Then as Vi :::; V n VB by hypothesis, we conclude [V, VB] = V2. This

completes the proof. D

13.5.2. The treatment of the subcase (Vf^1 ) nonabelian. We come to

the main result of the section, which determines the groups where (Vs°^1 ) is non-

abelian:

THEOREM 13.5.12. Assume Hypothesis 13.3.1 with L/CL(V) ~ An for n = 5

or 6, G ~ Sp6(2), and (V 3 °^1 ) nonabelian. Then either

(1) n = 5 and G ~ U4(2) or L4(3).

(2) n = 6 and G ~ U4(3).

The remainder of this section is devoted to the proof of Theorem 13.5.12.

Observe since G ~ Sp 6 (2) that Hypothesis 13.5.1 holds. Thus we may apply

results from earlier in the section; in particular by 13.5.7, we may apply results

from section F.9, and continue to use the conventions of Notation 13.5.8.

In the remainder of the section we assume (V 3 °^1 ) is nonabelian. Thus as G1 E

1iz, there exists HE 1iz such that UH is nonabelian.