15.1. INITIAL REDUCTIONS WHEN .Cr(G, T) IS EMPTY 109915.1.17, so by Remark C.1.19 we may take C 2 (S) = C2(T) and C 1 (T) ::::; C1(S).
Thus Nc(C2(S)) ::::; M by 15.1.17, while Cc(C1(S)) ::::; Mc since C 1 (T) ::::; Zand
Mc= !M(Cc(Z)). This establishes (*).
Next let GL := Ncl (L), ct== GL/Ccl (L), and zt := 01(Z(Ns(L)+))). We
establish:
(**) If 1zt1 = 2, then z+ = zt and CL+(Z+) = CL(z)+ ::::; (Mz n L)+.
Further if 0
31
(Mn L) f:. ML, then Cs(0
31
(Mn L)) = Cs(L).
For assume the hypotheses of(**). As L f:. Mz but Mc = !M(Cc(Z)), L does
not centralize Z, and hence z+ i=- 1. Therefore since IZtl = 2, z+ = zt, so
02 (CL+(Zt)) = 02 (CL(z)+) by Coprime Action. Then as the Sylow 2-group S
of G1 centralizes Z, CL+(Zt) = CL(z)+ ::::; (Mz n GL)+. Therefore if 031 (Mn
L) 1:. ML, then 031 (Mn L)+ does not centralize zt. However, Ns(L) acts onD := Cs(0
31
(Mn L)), so if D+ i=-1, then 1 i=-zt n D+. Then as IZtl = 2, zt
lies in D+, and so centralizes 031 ( M n L) +, contrary to the previous remark. This
completes the proof of(**).
Our final preliminary result says:
(!) If Po ~ P with (Po) 1:. ML, then P::::; M for each P E Po with P 1:. ML. If
in addition IZtl = 2, then Cs(0
31
(Mn L)) = Cs(L).Under the hypothesis of(!), the first statement follows from(*), and so in particular
03 ' (Mn L) f:. ML. Then the second statement follows from (**).
We now begin to show that one of the conclusions of the lemma must hold. By
15.1.21, cases (2a), (3a), and (3b) of 15.1.20 do not hold.Suppose that case (2b) or (3c) of 15.1.20 holds, but L is not an A1-block.
Therefore either Lis a block of type L3(2), A 6 , or G2(2), or L ~ G2(2)',^2 F4(2),
or^3 D 4 (2). In each case Ns(L) is trivial on the Dynkin diagram of L/0 2 (L); when
Lis a block, this follows since U(L)/Cu(L)(L) is the natural module. Thus eachminimal parabolic over SL is Ns(L)-invariant. Further in each case, CL(Zt) is
one of these minimal parabolics, with ML= CL(Zt) by 15.1.20. Let P denote the
other minimal parabolic over SL, and set Po := {P}. As 02 (P) is not an Arblock,
Po~ P. Thus if we can show that IZtl = 2, then conclusion (2) or (4) of 15.1.22
will hold by (!). When L+ is simple, this is a well-known fact (cf. 16.1.4 and 16.1.5)
about the structure of Aut(L), so we may assume L is a block. Here F*(L+) =
02 (L)+ = 02 (L)+ by A.1.8, so also F*(L+Ns(L)+) = 02(L+Ns(L)+) =:Qt, and
hence zt::::; Qt. But then zt = Cu(L)+(Ns(L)+) by Gaschiitz's Theorem A.1.39.
Thus IZtl = 2 from the action of Lon U(L), completing the proof that the lemma
holds in this case.
Next we consider the remaining case in (2b) of 15.1.20, where L is an A1-
block. Here we adopt the notation of section B.3, let P denote the preimage of
the stabilizer of the partition { {1, 2}, {3, 4}, {5, 6}, {7} }, and set Po := {P}. Again
Po ~ P. Further by 15.1.20, ML is the stabilizer of the vector e5,6 of U(L), and
hence P f:. ML, so P ::::; M by (!), completing the proof that conclusion (1) holds
in this case.
Now assume that case (2c) or (3e) of 15.1.20 holds, so that L/02(L) ~ L4(2)
or L 5 (2). Then S = Ns(L) by 1.2.1.3. Let Pc denote the parabolic generated bythe minimal parabolics for the interior nodes in the diagram for L / 02 ( L). In case
(3e), IZtl = 2, and by 15.1.20, ML is a proper parabolic containing Pc= CL(Zt).