494 INTRODUCTION TO VOLUME II
FSU where Vis not an FF-module. The only quasithin example which arises from
those cases is J4, but shadows of groups like the Fischer groups and Conway groups
complicate the analysis, and are only eliminated rather indirectly because they
are not quasithin. When Vis not an FF-module, the pair L/02(L), Vis usuallysufficiently far from pairs in examples or shadows, that the pair can be eliminated
by comparing various paramters from the theory of weak closure.
0,3.4.2. The Generic Case. In the Generic Case, L ~ L2(2n) and n(H) > 1
for some HE 7i*(T, M). We prove in Theorem 5.2.3 that the Generic Case leads
to the bulk of the groups of Lie type and characteristic 2 in the conclusion of our
Main Theorem; to be precise, one of the following holds:
(1) Vis the A 5 -module for L/02(L) ~ L2(4).
(2) G ~ M23.
(3) G is Lie type of Lie rank 2 and characteristic 2.To prove Theorem 5.2.3, we proceed by showing that if neither (1) nor (2) holds,
and D is a Cartan subgroup of L, then the amalgama:= (LTB,BDT,HD)
is a weak EN-pair of rank 2 in the sense of the "Green Book" [DGS85]; then by
Theorem A of the Gr~enBook and results of Goldschmidt [Gol80] and Fan [Fan86],
the amalgam a is determined up to isomorphism. At this point there is still work
to be done, as this determines G only up to "local isomorphism". Fortunately there
is a reasonably elegant argument to complete the final identification of G as a group
of Lie type and characteristic 2; this argument is discussed in the Introduction to
Volume I, in section 0.12 on recognition theorems. It also requires the extension
4.3.2 of Theorem 4.2.13 to show that G = (L, H).After dealing with the Generic Case, we still have to consider the situation
where L/0 2 (L) ~ L 2 (2n) and n(H) = 1 for all HE 7i*(T, M); in Theorem 6.2.20,
we show that then either Vis the A 5 -module for L/02(L), or G ~ M 22. Thus from
now on, if L/0 2 (L) ~ L 2 (2n), we may assume n = 2 and Vis the A 5 -module.0.3.4.3. Other FF-modules. Next in Theorem 11.0.1, we eliminate the cases
where L is SL 3 (2n), Sp 4 (2n), or G 2 (2n) for n > 1. From the list in section 3.2,
this leaves the cases where L is essentially a group of Lie type defined over F 2 ;
that is, Lis Ln(2), n = 3,4, 5; An, n = 5, 6; or U 3 (3) = G 2 (2)'-and Vis an FF-module. Roughly speaking, these cases, together with certain cases where Ct(G, T)
is empty, are the cases left untreated in Mason's unpublished preprint. They are
also the most difficult cases to eliminate.
We first show either that there is z E Zn V# with Gz := Ca(z) 1:. M, orG is As, Ag, M22, M23, M24, or L 5 (2). In the latter case the groups appear as
conclusions in our Main Theorem, so we may now assume the former.
Let Gz := Gz/(z), Lz := 02 (CL(z)), and Vz := Wlz), where \,12 is the preimage
of C-v-(T), and U := (V;,,°z). Then U ~ Z(02(Gz)) by B.2.14, and our next task is to
reduce to the case where U is elementary abelian. If not, then U = Z (U)Qu, whereQu is an extraspecial 2-group, and then to analyze Gz, we can use some of the
ideas from the the theory of groups with a large extraspecial 2-group (cf. [Smi80])in the original CFSG: We first show that if Z(U) -=/= (z/, then G ~ Sp 6 (2) or HS.
Hence we may assume in the remainder of this case that U is extraspecial. Then we
repeat some of the elementary steps in Timmesfeld's analysis in [Tim78], followed