1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

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906 13. MID-SIZE GROUPS OVER F2


PROOF. Assume Gz is solvable. Then using B.2.14 as usual, the pair H := Gz,

VH := (Z^0 z) satisfy the hypotheses of 13.4.8. Therefore by 13.4.8.3, H = IL1T,

where I:= (J(R 1 )H) arid H/0 2 (H) ~ 83 x 83. Also M = LTCM(V) = LCM(z) =
L(H n M) and H n M = L 1 T, so M =LT.


We next check next that the hypotheses of Proposition 13.4.7 are satisfied with

IT, L 2 T in the roles of "H 1 , H 2 ": For example, L 2 has at least three noncentral

2-chief factors, two on V and one on 02 (12), giving (a). Further ZL = Cz(L2T)

is of index 2 in Z by 13.4.3.2; while Cz(IT) is of index 2 in Z as m([VH,I]) = 2

by 13.4.8.2, and VH = [VH,I]Cz(I) by B.2.14, so that (b) holds. Suppose Ho:=
(IT, L 2 T) E 1i(T). As H = IL1T i M, I i M, so Ho i M = !M(LT) and
hence L i H 0. But L 2 ::; Ho and L2T is maximal in LT= M, so L2 = 0
31


(Hon

L) = 031 (Hon M) since L/02(L) ~ A 6 by 13.4.8.1. Hence L1 n Ho = 02(L1).

Further ZL = Cz(L2) and Ca(ZL) ::; M, so L2 = 0
31
(CH 0 (ZL))) :'.::l CH 0 (Z£).
As Gz = H = IL1T with L1 n Ho = 02(L1), 02 (1) = 0
31
(CH 0 (z)) :'.::l CH 0 (z),


and CH 0 (Cz(I))::; CH 0 (z). Hence (c) holds. This completes the verification of the

hypotheses of Proposition 13.4. 7.


Now by 13.4.7.1, Ho E 1i(T) and m(Z) = 2. Therefore m(VH) = 3 as z (j.

[VH,I] ~ E 4. Furthermore one of the cases (i)-(iv) holds. As L2 = 02 (H2),
conclusion (iii) is ruled out by 13.4.7.3, and conclusion (iv) is ruled out by 13.4.7.4.
If [0^2 (I),L 2 ] ::; 02 (0^2 (I)), then LT = (L 1 T,L 2 T) ::; Na(0^2 (I)), contrary to
I i M = !M(LT); this rules out conclusion (i). Thus Ho satisfies conclusion (ii),
and so Ho/02(Ho) ~ L3(2).


Let E 0 := M, E 1 := H, E2 :=Ho, :F := {Eo,E1,E2}, and E := (F). We show

that (E, :F) is a C 3 -system as defined in section I.5. First hypothesis (D5) holds as

Zv ::; Z(E 0 ). By 13.4.8.1, E 0 /0 2 (Eo) ~ A6 or 85, verifying hypothesis (Dl). We


have already observed that hypothesis (D2) holds, and hypothesis (D3) holds by

construction. Finally as ME M and Hi M, kerT(E) = 1, so hypothesis (D4) is
satisfied.
As (E, :F) is a C 3 -system, E ~ 8p 6 (2) by Theorem I.5.1. Thus it remains to
show that E = G. To do so we appeal to a fairly deep result on groups disconnected


at the prime 2, which we used earlier in our appeal to Goldschmidt's Theorem in

chapter 2. Let W := 02(E 2 ); as E ~ 8p5(2), W is the core of the permutation

module for E2/W and W = J(T). Thus H.5.3.4 tells us that E2 has four orbits
(3 1 , a 2 , "(2, (33 on W#, consisting of vectors of weights 6, 4, 2, 4, and the orbits
have length 7, 7, 21, 28, respectively. As W = J(T), E 2 controls G-fusion in W by


Burnside's Fusion Lemma A.1.35. As E ~ 8p 6 (2), it follows from [AS76a] that E

has four classes of involutions, determined by the Suzuki type of each on the natural


module-so these orbits contain representatives for the classes, namely the Suzuki

types b 1 , a 2 , c 2 , b 3 suggested by the notation above. Hence E controls G-fusion of

its involutions. As M = Ca(Zv) ::; E, it follows that E is the unique fixed point

on G/E of a generator d of Zv. For j tt (33, we may choose TE 8yb(CE 2 (j)), so
that F*(Ca(j)) = 02 (Ca(j)) by 1.1.4.6; hence d E 02(Ca(j)), so Eis the unique


fixed point of 02 (Cc(j)) on G/E, and hence Ca(j)::; E.

Set D := d^0. We claim that D is product-disconnected in G with respect

to E, in the sense of Definition ZD on page 20 of [GLS99]; cf. the proof of


I.8.2. Condition (a) of that definition is trivial. Since E ~ 8p 6 (2) we check that

dE n T = b1 n T = f31. Since E controls G-fusion of its involutions, D n E = dE,


while by the previous paragraph, Ca(d) ::; E. Thus condition (b) of the definition
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