1094 15. THE CASE .Cr(G, T) =^0
LEMMA 15.1.20. Let LE £(G 1 ,S) with L/0 2 (L) quasisimple and L -f:_ Mz.
Set SL:= Sn Land ML:= Mz n L. Then SL E Syb(L) and
(1) Li M.
(2) Assume F*(L) = 02 (£). Then L = [L, J(S)], and one of the following
holds:
(a) Lis a block of type A 5 or L 2 (2n), and ML is a Borel subgroup of L.
(b) L/02,z(L) ~ A1, £3(2), A5, or G2(2)'. Further if LE C(G1) then L
is a block of type A 7 , £ 3 (2), A 6 , or G 2 (2). In the last three cases, ML= CL(Z) is
the maximal parabolic subgroup of L centralizing Cu(L)(SL), and in the first case
ML is the stabilizer of the vector of U(L) of weight 2 centralized by BL.
(c) L/0 2 (£) ~ £ 4 (2) or £5(2), and ML is a proper parabolic subgroup of
L.
(3) Assume L is a component of G 1. Then Z(L) is a 2-group, and one of the
following holds:
(a) Lis a Bender group or L/0 2 (L) ~ Sz(8), and ML is a Borel subgroup
of L.
{b) L ~ L3(2n) or Sp4(2n), n > 1, or L/02(L) ~ £3(4), and ML is a
Borel subgroup or a maximal parabolic of L.
(c) L ~ G2(2)',^2 F4(2)', or^3 D4(2), and ML= CL(Z(SL)).
(d) L/02(L) is a Mathieu group, J2, HS, He, or Ru, andML = CL(Z(SL)).
(e) L ~ £4(2) or £5(2), and ML is a proper parabolic subgroup containing
CL(Z(SL)).
PROOF. If L ~ M, then by 14.1.6.1 and 15.1.5.2, L ~ CM(V) ~Mc, contrary
to the choice of L; so (1) holds.
Since SE Syb(G 1 ) by 15.1.18.4, and LE £(G 1 , S) by hypothesis, SL= Sn L
is Sylow in the subnormal subgroup L of (L, S).
Suppose that L/0 2 (£) is a simple Bender group; we claim that ML is the
Borel subgroup BL of L over SL. For BL is the unique maximal overgroup of
SL in L, so as SL ~ ML < L, it follows that ML ~ BL; then NMz(L) acts on
NL(0 2 (ML)) =BL, and hence Mz normalizes the Borel subgroup Bo := (Bs) of
Lo := (Ls). As L/02(L) is a Bender group, F*(Bo) = 02(B 0 ), so Bo ~ Mz by
15.1.19.2, and hence ML= BL, as claimed.
We now begin the proof of (2), so assume that F*(L) = 02 (£). Set H :=
LSH, where SH := Ns(L). Then SH E Syb(H) and as F*(L) = 02(L), also
F(H) = 02(H). Let U := (ZH) and H := H/CH(U), so that 02 (H*) = 1 by
B.2.14. As CH(U) ~ CH(Z) ~ Mz and L i_ Mz, L =j=. 1; thus as L/0 2 (£) is
quasisimple, L is quasisimple. As Nc(J(S)) ~ M by 15.1.17, and Li M by (1),
J(S) i 02( (L, S)) using B.2.3.3. Now by B.2.5, we may apply B.1.5.4 to conclude
that J(S) normalizes L, so that L = [L, J(S)]. Thus U is an FF-module for H*
by B.2.7. Therefore by Theorem B.4.2, L* is one of L 2 (2n), SL 3 (2n), Sp 4 (2n)',
G2(2n)', Ln(2), for suitable n, or A5, or A1.
Suppose L ~ L 2 (2n). If Lis a block then L/0 2 (£) ~ L 2 (2n), so Ml, is a Borel
subgroup of L by paragraph three, and hence conclusion (a) of (2) holds. So we
assume Lis not a block, and it remains to derive a contradiction. Now as L/0 2 (£)
is quasisimple, His a minimal parabolic, so we may apply C.1.26 to conclude that
either C1(SH) centralizes L, or C 2 (SH) :::l H. By 15.1.17, Baum(T) = Baum(S),
and by C.1.16.3, Baum(S) acts on L, and hence Baum(S) = Baum(SH) by B.2.3.4.