866 13. MID-SIZE GROUPS OVER F2
(1) Hypothesis 12.2.3 holds.
(2) Ca(v) i M for some v EV#.(3) L/02(L) ~ A5, A6, A6, Ls(2), or G2(2)'.
(4) If L/02(L) £:! A.6, then V/Cv(L) is the natural module for A6·
(5) If L1 E .C(G,T) and L1 :S: L, then L1 =LE .l*(G,T).PROOF. As L E .Cj(G, T) with L/0 2 (L) quasisimple, part (1) of Hypothesis
13.1.1 implies that Hypothesis 12.2.l holds, allowing us to apply Theorem 12.2.2.
Parts (2) and (3) of Hypothesis 13.1.1 exclude the groups in conclusions (1) and (2)
of Theorem 12.2.2, so that conclusion (3) of that result holds. Hence T normalizes L
and Hypothesis 12.2.3 holds, establishing (1). Part (3) of Hypothesis 13.1.1 excludes
the groups in conclusions (2)-(4) of Theorem 12.2.13, as well as the groups in the
conclusions of Theorems 12.3.1, 12.7.1, and 12.9.1. Hence those results eliminate
the corresponding cases from conclusion (3) in Theorem 12.2.2 and so establish
(2)-(4). Finally as the groups in (3) are of Lie type and either of Lie rank 2 over
F 2 , or A 5 of Lie rank 1, each proper T-invariant subgroup of Lis solvable. Then
(5) follows from 1.2.4. D
Define
.C+(G,T) :={LE .Ct(G,T): L/02(£) is notquasisimple },and suppose for the moment that .C+(G, T) is empty. If K E .Ct(G, T) then by
1.2.9, K::::; LE .Cj(G, T). As .C+(G, T) = 0, L/02(L) is quasisimple, so K =LE
.Cj(G,T) by 13.1.2.5. That is, once we show that .C+(G,T) is empty, we will be
able to conclude that .Ct(G, T) = .Cj(G, T).
REMARK 13.1.3. Recall that non-quasisimple C-components are allowed by the
general quasithin hypothesis: they appear as cases (3) and (4) of A.3.6, and cases
(c) and (d) of 1.2.1.4. On the other hand, they do not actually arise in .Ct(G, T) in
any of the groups in our Main Theorem. Thus after Theorem 13.1.7, we will finally
be rid of this nuisance. In particular, if .Cj(G, T) is nonempty, then by 3.2.3 there
will exist tuples in the Fundamental Setup. Furthermore, as we just observed, we
will also have .Ct(G,T) = .Cj(G,T).
If LE .C+(G,T) then L appears in case (c) or (d) of 1.2.1.4, so mp(L) = 2 for
some odd prime p dividing the order of 02 ,F(L), and T::::; No(L) by 1.2.1.3. Also
in the notation of chapter 1, 1 'I 3p(L) E 3( G, T) by 1.3.3.
Recall the basic facts about 3( G, T) from that chapter. Recall also from Defi-
nition 3.2.12 that 3_(G, T) consists of those X E 3(G, T) such that either Xis a
{2, 3}-group, or X/0 2 (X) is a 5-group and Auta(X/0 2 (X)) is a 2-group. Further
3+(G, T) is defined to be 3(G, T) - 3_(G, T). Set
2+(a, T) := 3+(G, T) n 3*(G, T).
We will make repeated use of results from section A.4 such as A.4.11.
We next collect some useful properties of the members L of .C+(G, T). Although
the proof of the next lemma contains an appeal to 13.1.2.3, we could in fact have
stated and proved 13.1.4 much earlier, after chapter 11. On the other hand, manyarguments from now on (eg. the proof of 13.1.9.1) make strong use of 13.1.2.5-
which does depend on work done in chapters after chapter 11.