882 i3. MID-SIZE GROUPS OVER F2
that lemmas= 2 and E = (ei, e 2 ), where ei = ei,2 and e2 = e3,4 are nonsingular.
Further T 0 = CT(V2) E 8yh(Ca(e2)) by 13.2.6.1. Set Ki := 02 (Hi), K2 :=
02 (CL(e 2 )), Gi := KiTo, and Go:= (Gi, G2)· Then Go::::; Ca(e2), so in particular
To E 8yh(G 0 ) and Go is an SQTK-group. Therefore (Go, Gi, G2) is a Goldschmidt
triple of Definition F.6.1 in section F.6, so we can appeal to results in that section.
Let X := 031(Go), Go:= Go/X, a:= (G1, To, G2), and Qi:= 02(Gi). Observe
that K 2 = ((1,2), (1,5)) and Q 2 = ((3,4)). Further Xis 2-closed by F.6.11.1.
Suppose first that Qi= Q 2. By Theorem 4.3.2, M = !M(L), so as Ki i M, no
nontrivial characteristic subgroup of Q2 is normal in LQ2. On the other hand the
hypotheses of C.1.24 are satisfied with Q 2 in the role of "R", so Lis an As-block
by C.1.24, contrary to 13.2.11.5.
Therefore Qi -=J-Q 2. In particular a is a Goldschmidt amalgam by F.6.11, so as
G 0 is an SQTK-group, Go is described in Theorem F.6.18. Further by the previous
paragraph, case (1) of F.6.18 does not arise.Suppose next that ei E 02(Go). Then W := (ef^0 )::::; 02(Go). As the generator
z := eie2 of Z lies in W(e2), Na(W(e2)) E 1-le by 1.1.4.3, and hence A:= Na(W)n
Ca(e 2 ) E 1-le by 1.1.3.2. Then as To E 8yh(A) since To E 8yl 2 (Ca(e 2 )) and
To ::::; Go ::::; A, we conclude Go E 1-le by 1.1.4.4. As [Ki, ei] -=J-1, Ca; (W) ::::; Qi for
i = 1, 2, so Ca 0 (W) is 2-closed and solvable by F.6.8. Further as To E 8yl2(Go)
and ei E Z(To), WE R2(Go) by B.2.13. As K 2 = ((1, 2), (1, 5)), it follows from
13.2.11.2 that K2 = [K2, J(T)] and J(T) = J(To). By 13.2.8.4, Ki = [Ki, J(T)].
Therefore Wis an FF-module for G 0 := Go/Ca 0 (W) with Kt =[Kt, J(To)*] #-1.
Assume first that Go satisfies one of conclusions (3)-(13) of F.6.18, and let
Lo := G 0 and Wo := [W,L 0 ]. Then from F.6.18, Lo is quasisimple, so as X
is 2-closed, Lo E C(Go) by A.3.3. As we saw Go E 1-le, L 0 E 1-le by 1.1.3.1,
so that LoTo E 1-le. By F.6.18, Lo contains Ki or K 2 ; so as Ki = [Ki, J(T)],
Lo = [Lo, J(To)]. Hence LQJ(T)* is described in Theorem B.5.1. Comparing that
list with the list in F.6.18, we conclude that Lo~ L 3 (2), 8p 4 (2)', G 2 (2)', or A1, and
Wo/Cw 0 (Lo) is a natural module for L 0 , a 4-dimensional module for L 0 ~ A 7 , or
the sum of two isomorphic natural modules for L 0 ~ L 3 (2). In each case F.6.18 saysLo = 02 (G 0 ), so Ki ::::; Lo. Then the condition that neither Ki nor K 2 centralizes
ei E Cw (To) eliminates all cases except the one where W 0 is the natural module
for G 0 ~ 81 and (in the notation of section B.3) for i := 1 or 2, Gi is the stabilizer
of a partition of type 22 , 3, while G3-i is the stabilizer of a partition of type 23 , 1.
This is impossible, as in that case J(T)* = 02 (G3-i), contrary to Kj = [Kj, J(T)]
for each j.This contradiction shows that Go satisfies none of conclusions (3)-(13) of
F.6.18; as case (1) of F.6.18 was eliminated earlier, we conclude that case (2)
of F.6.18 holds. Therefore G 0 ~ 83 x 83 or E 4 /3i+^2. As W is an FF-module
for G 0 and Ki = [Ki, J(T)] for i = 1 and 2, it follows from Theorem B.5.6 that
Kt ::::! G 0 ~ L2(2) x L2(2), and W = [W,Ki] EB [W,K2], with [W,Ki] ~ E 4. Re-- K