1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

13.8. FINISHING THE TREATMENT OF A 6 949

Suppose one of the first three cases holds, namely UH is an irreducible module.

To eliminate these cases, it will suffice to show:

v.r ::; V2 for some h E H. ( *)

For if() holds, then Vt= V{ for l E LzT with l^2 EH. As G1 = H, UH :':1 G 1 ,
so as Vt= V{, also U~ = Ufih = Uit-. Thus as l^2 EH, l interchanges UH and U~,
and also QH and Q~, impossible as Ua i QH but UH ::; Qa. This completes the
proof of the sufficiency of(). Now we establish () in each of the first three cases:
If UH is the L4(2)-module or 86 -module, then() holds as His transitive on tif;.
If UH is the 87-module, then() follows from 13.7.6.3b, which says V 2 is of weight
2, using the fact that the center Vf of the transvection u; is of weight 2.
Thus we may assume that UH is a 5-dimensional module for H ~ 86. As u;
induces transvections on UH with center.Ai, u; has order 2, so Da := UanQH is a
hyperplane of Ua; and as D 7 = U 7 , [Da, UH] = 1 by F.9.13.7. As u; =/= 1, without
loss V{* =/= 1 and [Vf, Vi]=/= 1. Thus as we saw [Ua, UH]= Vf, [Vf, Vi]= Vf::; Vf;
so Vi ::; Cc(Vf) n Nc(Vf) ::; Mt by 13.5.5. Then Vi lies in the unipotent radical
of the stabilizer in Mt of Vf, and is nontrivial on the hyperplane Vf orthogonal
to Vf, so [V^9 , Vi]> Vf.
Define C to the preimage in UH of CuH(Vf); then [Ua,C] ::; Vf n Vi = 1,
so C ::; H~ and hence [V9, C] ::; Vf by 13. 7.3. 7. Thus from the action of 86 on

the core of the permutation module, Vi = V9 is the group of transvections with

center A1, so V9 = Vf (V^9 n QH)· Now [V^9 n QH, Vil ::; V^9 n Vi = 1by13.8.3.
Thus [V9, Vi] = [Vi(V9 n QH), V3] = [Vf, Vi] = Vf, contrary to the previous
paragraph. D

LEMMA 13.8.8. Either:

(1) D 7 = U 7 or DH= UH, and U 0 or Vo induces a nontrivial group of transvec-

tions on UH, for r5 := 1gb"^1 or "f, respectively. Hence H =KT for some KE C(H),

and H* and its action on UH are described in 13. 8. 5 ..

(2) D 7 < U 7 , DH < UH, and we may choose 'Y so that 0 < m(U;) ~

m(UH/DH), and u; E Q(H*,UH)· Further there is h EH with 12h = "fo, and

setting a:= 1h, Va= v;::; 02(L1T) ::; Ri.

PROOF. If D 7 = U 7 , then (1) holds by F.9.16.1. Similarly as in the proof

of the previous lemma, (1) holds if DH = UH. Thus we may assume D 7 < U 7 ,

so by F.9.16.4, we may choose 'Y as in conclusion (2); then the final statement of

conclusion (2) follows from parts (1) and (2) of F.9.13. D

LEMMA 13.8.9. Assume some F::; UH is V 7 -invariant and G 7 = (FG^7 )G 7 m_ 1 •

Then

{1) [F, V 7 ] i U 7.

(2) If [F, U 7 ] = [F, V 7 ], then Vi i U 7 and F induces a group of transvections

on V 7 /U 7 with center V1U 7 /U 7.

PROOF. Assume [F, V 7 ] ::; U 7. Then F centralizes V 7 /U 7 , so X := (FG^7 )

does also. But by 13.8.4.5, V 7 b_ 1 U 7 /U 7 is of order 2, so the section is centralized

G
by G 7 m-u and hence also by G 7 = XG 7 m_ 1. But then as V 7 = (V 7 b2i), G 7

centralizes V 7 /U 7 , contrary to 13.7.7. Thus (1) is established.