1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

6.2. IDENTIFYING M22 VIA L2(4) ON THE NATURAL MODULE 683

such that K = [A, K], A S K, K* induces GL(U 1 ) on U 1 := (BK) of rank at

least 3, and the kernel of the action lies in 02 (K). But 02 (H) = 1 by 6.2.4.4, so

K* ~ GL(U1). Then by 6.2.4.6, K* ~ L 3 (2) (so that m(U 1 ) = 3) and K = 031 (H).

As [UH,A] =BS U1 and K = [K,A], U1 = [UH,K]. We saw m(UH/CuH(A)) = 2,

so UH= U1ffiCuH(K). (cf. B.4.8.3). Now Du oforder 3 in Lu acts on the subgroup

R of 6.2.4.2, and then on RK := Rn K in view of 1.2.1.3. But R'K E Syb(K*),

so R'K is self-normalizing in K* and hence [D[r, K*] = 1. Then Du centralizes U 1

since K* = Aut(U 1 ). As A* SK*, this contradicts our claim in paragraph two.

This contradiction shows that B n U = 1 and that

.,....-____-

Cu H ( A1) = CuH(A) for each 1 =!=A! SA*.

Since A*~ E 4 , (*) says

and since B n U = 1 we have

(!)

Further applying(*) when Ai= A and recalling m(UH/CuH(A)) = 2, we conclude.

(! !)

Thus A* is an offender on the FF-module UH. Recall UH E R2(H) by 6.2.4.4,

and let KA_ := (A*H). By (*) and E.4.1, A centralizes O(H*), so that F(KA.) S

Z(KA_). Next(!!) restricts the possible components K* of KA. in the list of Theorem

B.5.6 to alternating groups or groups defined over F 2 or F 4. Now K* is the image

of K E C(H), and by 6.2.4.6 and inspection of our restriced list from B.5.6, either

(i) m3(K) = 1, so that K ~ L2(4) or L3(2), or

(ii) K ~ SL3(4) and Du induces outer automorphisms on K.

In particular K* = J(KA.)^00 is described by Theorem B.5.1. As in the previous

paragraph, RK :=Rn K E Syb(K).

Suppose first that case (ii) holds. By Theorem B.5.1.1, either VK := [UH, K*] E

Irr +(K, UK), or VK is the sum of two isomorphic natural modules for K. In the

former case, VK is a natural module by B.4.2. In either case, A.3.19 contradicts

the fact that Du 1:. K.

Thus case (i) holds. By Theorem B.5.1.1, either VK := [UH, K] E Irr +(K, UH),

or K* ~ L 3 (2) and VK is the sum of two isomorphic natural modules.

Assume first that K ~ L3(2). If VK is the sum of two isomorphic natural
modules, then by (), A* induces the group of transvections with a fixed axis on

each of the natural summands, contrary to(!). Thus VK E Irr+(K*,UH)· Then

by B.4.8.4, VK = [UH, K*] is either the natural module or the extension in B.4.8.2.

Now as Du acts on R'K E Syl2(K), Du centralizes K and VK/Cv-K(K*), and

hence Du centralizes V K by Coprime Action. As A S K, this contradicts our

claim in paragraph two.
This contradiction shows K* ~ L 2 (4), so VK E Irr +(UH, K). Then by B.4.2,

either VK is the As-module, or VK /CvK (K) is the natural module. The first case

is impossible by (). Thus the second case holds, and A E Syb(K*) by B.4.2.1.

Further CvK (K) = 1 by (!),so VK is the natural module, and UH= VK ffi CuH (K)

by B.5.1.4.