1096 15. THE CASE .Cf(G, T) =^0some z E z# induces an inner automorphism on L, then as Mc= !M(Ca(Z)), we
have CL(Z(SL)) :::-;; CL(z) :::-;; ML. ·
We first treat cases (a)-(c) of 1.1.5.3, where L/0 2 (L) is of Lie type in char-
acteristic 2, and hence described in case (3) or (4) of Theorem C (A.2.3). AsML is contained in a proper overgroup of SL E Sylz(L), ML is contained in a
proper parabolic subgroup PL of L (cf. 47.7 in [Asc86a]). In cases (a)-(c) of
1.1.5.3, either L ~ A 6 , or Z induces inner automorphisms on L. In the latter case,
CL(Z(SL)) :::-;; PL by the previous paragraph, and in the former CL(SL) =SL:::-;; PL
trivially.
Next if L/0 2 (L) is of Lie rank 1, then by 1.1.5.3, L/02(L) is a simple Bender
group, and conclusion (a) of (3) holds by paragraph three. Thus we may assume
that L/0 2 (L) is of Lie rank at least 2. Then by 1.1.5.3, either L is simple, orL/02(L) ~ L3(4) or G2(4).
Assume first that L/0 2 (L) is defined over F2"' with n > 1. This rules out case
(4) of Theorem C, so that L/0 2 (L) is in one of the five families of groups of Lie
rank 2 in case (3) of Theorem C. Further L ::::! G1 by 1.2.1.3. If S is nontrivialon the Dynkin diagram of L, then either L ~ L 3 (2n) or Sp 4 (2n) with n > 1, or
L/0 2 (L) ~ L 3 (4); further the Borel subgroup B of L over SL is the unique S-
invariant proper parabolic subgroup of L containing SL, so arguing as in paragraph
three, ML= B, and then conclusion (b) of (3) holds.
Thus we may assume that S normalizes both maximal parabolics Pi, i = 1, 2,
of Lover SL. Then Li :=Pt) E ,C(G1, S) with F*(Li) = 02(Li), and Li/02(Li)
is either L 2 (2m) (with m a multiple of n) or Sz(2n). By a Frattini Argument,
Mz = MLNMz(SL), and NMz(SL) = 02 (NMz(SL))S acts on the two maximal
overgroups P 1 and P 2 of SL in L. Thus Mz acts on each parabolic P containing
ML, so 02 , 2 1(P) :::-;; Mz by 15.1.19.2. Then if Li:::-;; Mz, Pi:::-;; Mz by 15.1.19.4, so
Pi= ML by maximality of Pi.
If Li is not a block, then Li :::-;; Mz by (2). Thus if neither L 1 nor L2 is ablock, then L = (L 1 , L 2 ) :::-;; Mz, contrary to hypothesis. Therefore Li is a block for
i = 1 or 2, so either Lis L3(2n) or Sp4(2n), or L/02(L) ~ L3(4). So if Ll :::-;; ML,
then Mz = P 1 by the previous paragraph, so that conclusion (b) of (3) holds.Thus we may assume that neither Ll nor L 2 is contained in ML. Then (cf. 4 7. 7 in
[Asc86a]), ML is contained in the Borel subgroup P1 nP2 = P over SL, so ML= P
by the previous paragraph, and again conclusion (b) of (3) holds.Thus we may assume that L/02(L) is defined over F 2. Then from 1.1.5.3,
and recalling Z(L) is a 2-group, L is simple. So from Theorem C, L is G 2 (2)',
(^2) F4(2)', (^3) D4(2), Sp4(2)', L
3 (2), L4(2) or L 5 (2). Recall from earlier discussion that
Pc:= CL(Z(SL)) :::-;; ML:::-;; PL for some proper parabolic PL of L. However in the
first three cases, Pc is a maximal parabolic, so ML= Pc, and hence conclusion (c)of (3) holds. Thus we may assume one of the remaining four cases holds. In those
cases, all overgroups of SL are parabolics, so ML is a parabolic. Thus conclusion
(e) of (3) holds if L ~ L4(2) or L5(2).In cases (a)-(c) of 1.1.5.3, we have reduced to L ~ L 3 (2) or A 6. We now
eliminate these cases, along with case (d) of 1.1.5.3. Since Z(L) = 02 (L), L ~ A1
in the last case. In each case SL~ D 8 , and Z(SL) is of order 2.
We claim that Z contains a nontrivial subgroup ZL inducing inner automor-
phisms on L. If L ~ L3(2), this follows from earlier discussion. In the other two
cases, Lis normal in G1 by 1.2.1.3, so as Out(L/02(L)) is an elementary abelian