1098 15. THE CASE Cr(G, T) = 0
(ii) 02 (B) = [0^2 (B), J(S)], and either L ~ L 2 (4) or Lis an A5-block.
(iii) L ~ U 3 (2n), some A E A(S) does not induce inner automorphisms on L,
and J(S) ::9 DS, where Dis the subgroup of B generated by all elements of order
dividing 2n - 1.
Suppose that case (i) or (iii) holds. Let Bo := B in case (i), and Bo := D
in case (iii). By 15.1.17, J(T) = J(S) and Bo :::; Na(J(S)) :::; M. As B :::; Mz,
Bo:::; Mn Mz =Mn Mc n G1, so by 15.1.18.2, Bo= 02 (B 0 ) centralizes V. Thus
112 :::; Cs(Bo) :::; Cs(L) from the structure of Aut(L) in (i) or (iii), contrary to
15.1.19.7.
Therefore case (ii) holds. Now J(T) = J(Cr(V)) by 15.1.9.1, so that M =
CM(V)NM(J(T)) by a Frattini Argument. Hence by construction of Z1 and 112 in
15.1.16, there is a p-subgroup Y of NM(J(T)) n G1 where p := 3 or 5, satisfying
SY = YS and 112 = [112, Y]. Now YS = SY, Y acts on J(T) = J(S), and
02 (B) = [0^2 (B), J(S)], so it follows from the structure of Aut(Lo) that [L, Y] = 1.
But then 112 = [112, Y] centralizes L 0 , contrary to 15.1.19.7. This contradiction
completes the proof. D
We now define notation in force for the remainder of this section: By 15.1.19.6,we may choose L E C(G 1 ) with L/02(L) quasisimple and L 1:. Mz. Set ML :=
Mz n Land SL:= Sn L. Then Lis described in 15.1.20. In the next lemma, we
refine that description.
LEMMA 15.1.22. One of the following holds:
(1) Lis an A1-block. Further L = (MnL, ML), MnL is a proper subgroup of
L containing the stabilizer of the partition of type 23 , 1 stabilized by SL, and ML is
the stabilizer of the vector of weight 2 in Cu(L) (SL)·
(2) L is a block of type L3(2), A6, or G 2 (2), ML is the maximal parabolic
subgroup of L centralizing Z, Mn L is the remaining maximal parabolic over SL,
and 08 (0
31
(Mn L)) = Cs(L).
(3) F*(L) = 02(L) with L/02(L) ~ L4(2) or L5(2), and ML and Mn L are
proper parabolic subgroups of L which generate L.
(4) L ~ G2(2)',^2 F4(2)', or^3 D4(2), ML= CL(Z(SL)), Mn L is the remaining
maximal parabolic over SL, and Cs(0
31
(Mn L)) = C 8 (L).
(5) L/Z(L) is J 2 , He, or a Mathieu group other than M 11 , ML= CL(Z(SL)),
and Cs(0
31
(Mn L)) ~ Cs(L).
(6) L ~Mn, ML= CL(Z(SL)), and 02 (Na 1 (J(S))):::; Ca 1 (L).
(1) L ~ L4(2) or L5(2), L = (ML,M n L), where ML is a proper parabolic
containing CL(Z(SL)), MnL is a proper parabolic, and Cs(0
31
(MnL)) = Cs(L).
PROOF. Observe that Mn L < L by 15.1.20.1, and ML < L since L 1:. Mz by
the choice of L. By 15.1.19.1, Z(L) is a 2-group.
We first establish some preliminary technical results. The first is on overgroupsof SL in L. Let P be the set of Ns(L)-invariant subgroups P of L such that
F*(P) = 02 (P), PNs(L)/02(PNs(L)) ~ S 3 or S 3 wr Z 2 , and 02 (P) is not a
product of A 3 -blocks. We show:
(*)For PEP, either P:::; MnL or P:::; ML.