S22
0- PUSHING UP IN QTKE-GROUPS
we chose TB E Syl 2 (MB), TB is transitive on each orbit of MB on parabolics of K
containing TB n K, and hence MK is a maximal TB-invariant parabolic.
As LB is a nonsolvable subgroup of MK, K is of Lie rank at least 2, and MK
is of Lie rank at least 1. Assume that K is of Lie rank exactly 2. Then as MK is
a proper parabolic of rank at least 1, it must be of rank exactly 1, and hence is a
maximal parabolic. Also LB = M'j( as M'j( /0 2 (MK)^00 is quasisimple. Then as
VB ::; Z(02(LB)) and VB= [VB,LB] we conclude by inspection of the parabolics
of the rank 2 groups that M+/0 2 (M+) 9! LB/0 2 (LB) 9! L2(2n), and either VB is
an FF-module, or (when K is unitary) VB is the n4(2n/^2 )-module for LB/02(LB)·
These are cases (i) and (ii) of conclusion (2) in Theorem 4.4.3, ;:i,nd in case (i) there
are no further restrictions on VB since L/0 2 (L) 9! L 2 (2n). This contradicts the
choice of B as a counterexample to Theorem 4.4.3.
Therefore K is of Lie rank at least 3, so as we saw from Theorem B, K 9! L4(2),
L5(2), or Sps(2). Thus M+/02(M+) 9! LB/0 2 (LB) 9! La(2), L4(2), or As, and
either VB is an FF-module, which is a natural module in the first two cases, or
K 9! Sps(2), LB/0 2 (LB) 9! La(2), and VB= 02 (LB) is the core of a 7-dimensional
permutation module for LB/0 2 (LB)· But then case (i) or (iii) of Theorem 4.4.3.2
holds, contrary to the choice of B as a counterexample, and completing the proof
of 4.4.12. D
We are now in a position to complete the proof of Theorem 4.4.3.
By 4.4.12, K/Z(K) is not of Lie type and characteristic 2. By 4.4.10.2, VB::; K.Assume first that m(VB) ::; 4. Then inspecting the list of quasisimple subgroups
of GL4(2), LB/02(LB) is one of L2(4), La(2), L4(2), As, or A7, with VB an FF-
module, or an A5-module for L2(4). Further if LB/02(LB) ~ La(2) or L4(2), then
either VB is a natural module for LB/0 2 (LB), so condition in (a) of subcase (i)
of case (2) of Theorem 4.4.3 is satisfied, or m(VB) = 4 and LB/02(LB) ~ La(2).
The former case contradicts our assumption that B is a counterexample, so we
may assume the latter holds. Then as VB = [VB, LB], ZB := GvB (LB) is of
rank 1 and VB/ZB is a natural module. By 4.4.6.1, MKTB = GKTB(ZB), so
LB SJ GK(ZB)· Examining involution centralizers in the groups appearing in
Theorem B for such a normal subgroup, we conclude K 9! M 23 ; but there LB is
not normal in NK(VB) 9! A7f E15.
Thus we may assume that m(VB) > 4, and hence m 2 (K) > 4. Then from the
list of Theorem B, K/Z(K) is not L2(Pe), L3(p), PSp4(p), L4(p), G2(p), A7, Ag, a
Mathieu group other than M 2 4, a Janka group other than J 4 , HS, or Mc.
Since K/Z(K) is not of Lie type and characteristic 2 by 4.4.12, we conclude
from Theorem B that K/Z(K) is M 24 , J 4 , He, and Ru. Since the multipliers of
these groups are 2-groups by l.1.3, while 02 (K) = 1 by 4.4.7.1, it follows that K is
simple. Again by 4.4.6.1, MKTB is the unique maximal 2-local subgroup of KTB
containing LBTB. Inspecting the maximal 2-locals of Aut(K) for a nonsolvable
2-local MKTB such that LB ~ MKTB and 1-=/= VB= [VB,LB]::; Z(02(LB)), we
conclude one of the following holds: