598 3. DETERMINING THE CASES FOR LE .Cj(G, T)Cz(L). By (**), 031 (Ca(zd)) = Ld and Ld i= Le since ILi > !Lei· Therefore by
(*), zd ¢. Cz(L)ee = Ze - Cz(L), and hence zd E Cz(L), so again using (**),
L = 031 (Ca(zd)) = Ld. Thus d E Na(L) = M by 1.4.1 in this case. In the
remaining case, z E Ze - Cz(L) = Cz(L)ee, where by (*), 0
31
(Ca(z)) =Le, and
hence 031 (Ca(zd)) =Lg -=/= L. Therefore by (*), zd E Ze - Cz(L) = Cz(L)ee,
and then Le = 031 (Ca(zd)) by (), and hence Le =Lg. Thus d normalizes LeT
and hence also 02 (LeT) = 02 (LT), so again d E M. Therefore D :::; M, contrary
to 3.3.6.a, establishing the claim.Next Cv(Z) :::; Cv(ee), and Cv(ee) normalizes 031 (Ca(ee)) = Le using (*),
and hence also normalizes 02 (LeT) = R. Therefore Cv(Z) :::; Na(R) :::; M as
C(G,R) :::; M, so Cv(Z) <Das Di M. As Z is of rank 2, we conclude ID:
D n Ml= 3, with D transitive on z#. In particular there is d ED with e~, 6 = ee.
- d 31
Let Ls,6 := CL(es,6)^00 • Thus Ls,6T ~ Z2 x S5, and L 5 , 6 :::; 0 (Ca(ee)) = Le.
This is impossible, as T = Td acts on L~ 6 and Le, whereas there is no T-invariant
subgroup of Le/ 02,z (Le) ~ A5 isomorphic to As.
We have shown that L is not A 7. Thus case (3) of 3.3.8 does not hold by
3.3.10.2. This completes the proof of 3.3.22. DNotice that at this point, cases (1), (2), (3), (5), and (7) of 3.3.8 have been
eliminated by 3.3.14, 3.3.10.2, 3.3.22, 3.3.17, and 3.3.21. Thus leaves case (6) of
3.3.8, where L ~ A 6 , and case (4) of 3.3.8, where L ~ Ls(2), A5, or Us(3) by 3.3.16.
In each of these cases, L/0 2 ,z(L) is of Lie type and Lie rank 2 in characteristic 2,and T normalizes L. Threfore by 3.3.6.d, (LT, T) is an MS-pair in the sense of
Definition C.1.31. Thus we may apply C.1.32 to LT to conclude that either Lis a
block, or L ~ L 3 (2) is described in C.1.34. We first investigate the latter possibility
in more detail:LEMMA 3.3.23. If L is L 3 (2), then either
(1) L is an L 3 (2)-block, and D acts on the preimage To in T of Z(T), or
(2) L has two or three noncentral 2-chief factors, and D does not act on02(CL(Z)T).
PROOF. As in earlier arguments we conclude that one of cases (1)-(4) of C.1.34
holds. In particular [V, L] is a sum of r :::; 2 isomorphic natural modules, so by
3.3.7.4, V = [V, L] E8 ZL and Z = (Zn [V, L]) E8 ZL, where Zn [V, L] has rank r.
Suppose case ( 4) of C.1.34 holds; we argue as in the proof of 3.3.18, although
many details are now easier: As M = !M(LT), M = !M(Ca(z)) for each z E zf,and in case (4) of C.1.34, m(ZL) :::: 2 and r = 1 so ZL is a hyperplane of Z, leading
to the same contradiction as in the proof of 3.3.18.Thus we are in case (m) of C.1.34 for some 1 ::::; m ::::; 3, where L has m
noncentral 2-chief factors. This gives the first statements of (1) and (2). Next in
each case of C.1.34, T::::; L0 2 (LT). Set X := 02 (CL(Z)) and R := 02 (XT). NowLR, R also satisfy (MSl) and (MS2), but if m = 2 or 3, then (LR, R) is not an
MS-pair as the corresponding cases of C.1.34 exclude this choice of R. Therefore
(MS3) must fail for R, so there is a nontrivial characteristic subgroup C of R normal
in LR, and hence normal in LT as R~T. Thus Na(R)::::; Na(C)::::; M = !M(LT),
so D does not act on Ras D i M by 3.3.6.a, proving the second statement in (2).