12.2. GROUPS OVER F2, AND THE CASE V A TI-SET IN G 799may apply A.3.15. However, the list of possibilities for L/0 2 (L) from A.3.15 does
not intersect the list from Theorem 12.2.2.3.Therefore B ::::; CMv(V) ::::; CMv(L), and hence B ::::; CMv(L/0 2 (L)). Visibly
Hypothesis 4.2.1 is satisfied, so Hypothesis 4.4.1 is satisfied by Remark 4.4.2. Thuswe can apply Theorem 4.4.3 to obtain a contradiction: Namely we showed that
Nc(A) 1:. M, so that one of the conclusions of 4.4.3.2 must hold, which is not the
case as we are assuming one of hypotheses (a)-(c). DThe statement of the following lemma makes use of Notation 12.2.5.
LEMMA 12.2.12. Assume v E V# with 02(Lv'f'v) = 1, and choose Tv E
Syl2(Mv)· Then(1) Q = 02(LvTv), Q E Syb(CcJLv/02(Lv))), and Hypothesis C.2.3 is sat-
isfied with Gv, Mv, Q in the roles of "H, MH, R".
(2) Assume Lv = L';: and V = [V, Lv]. Then Hypothesis C.2.8 is satisfied with
Gv, Mv, L';:, Q in the roles of "H, MH, LH, R".
PROOF. Since V E R.'2(LT), Q ::::; Tv, so as 02(Lv'f'v) = 1, Q = 02(LvTv) by1.4.1.4. We chose Tv E Syb(Mv) while Lv :::1 Mv and C(Gv, Q) ::::; Mv by 12.2.5.3.
Therefore (1) follows from A.4.2.7.
Assume Lv = L';: and V = [V, L';:]. Then L';: E C(Mv), and as Lv = L';:,
the argument in the previous paragraph shows that Q E Syb ( C Mv (L': / 02 ( L':))).Also Mv E He by (3c) of 12.2.5, and the verification of the remainder of Hypothesis
C.2.8 is straightforward. D12.2.2. The treatment of V a TI-set in G. We now come to the main
result of this section, in which we treat the case where V is a TI-set in G.THEOREM 12.2.13. Assume Hypothesis 12.2.3. Then one of the following holds:
(1) Cc(v) 1:. M for some v EV#.
(2) Lis an Ln(2)-block for n = 3 or 4, and G ~ Ln+i(2).
(3) L is an L 3 (2)-block, and G ~Ag.
(4) L is an L4(2)-block, and G ~ M24.REMARK 12.2.14. The groups M22 and M23 contain a pair (L, V) failing 12.2.13.1,
with V ofrank 4 and L/02(L) ~ A6 or A7, but these groups are explicitly excluded
by Hypothesis 12.2.3. Their shadows are eliminated via an appeal to 12.2. 7.3, which
is violated in M22 and M23.
REMARK 12.2.15. As the groups appearing in conclusions (2), (3), and (4) of
Theorem 12.2.13 appear as conclusions in our Main Theorem, we will sometimes
assume that G is not one of those groups. Then Theorem 12.2.13 tells us that
Cc(v) 1:. M for some v EV#.
Until the proof of Theorem 12.2.13 is complete, assume that Cc(v) ::::; M for
each v E V#. We must show that one of (2)-(4) holds. We begin a series of
reductions. Recall we have adopted Notation 12.2.5.
LEMMA 12.2.16. V is a TI-set in G.PROOF. By 12.2.6, V is a TI-set in M. Thus if the lemma fails, there is
g E G - Mand v EV# with u := v9 EV. As we are assuming that conclusion (1)
of 12.2.13 fails, Gw = Mw for w E V#, so