1138 15. THE CASE .Cr(G, T) = r/J
L. Now Y 1 centralizes Vz and hence also VL, whereas [VL, Y1] = [VL, X] # 1, a
contradiction. D
The next lemma rules out conclusion (1) of 15.3.37, and eliminates the shadowof G ~ S 9 where L ~ A 5 , and also those of G ~ L wr Z2 for L ~ L3(2) and A5.
LEMMA 15.3.40. Y+ '.S L.
PROOF. Assume Y+ 1:. L. Then m 3 (L) = 1by15.3.38, and L =Lo by 15.3.39,
so that case (2) of 15.3.37 holds, and in that notation of the lemma, Y+ = YLYe
with JYLl3 = 3 = JYeJ3, and Y+/02(Y+) ~ Eg. As Lis S-invariant, the subgroups
YL and Ye are S-invariant. By 15.3.36.1, Y+S/ R ~ S3 x S3, so from the structure
of Min 15.3.2.1, {Ye, YL} = {Y 1 , Y 2 }, where Y2::::; Y+ n M2 with·V2 = [V, Y2], and
Yi::::; Y+ nM1.
If Ye= Y 2 then Vz = [Vz, Ye]::::; Oc(L), so as L 1:. M, Oc(V2) 1:. M, and hence
Y = Y+ by 15.3.11.3. Then since V 1 = [Vi, Y 1 ], interchanging the roles of Vi and Vz
if necessary, we may assume instead that YL = Y2. As YL = Y2, V2 = [V2, YL]::::; L.
Suppose first that L ~ L 2 (2n) or U 3 (2n). Then Mr acts on the Borel subgroup
over SL, so ML is that Borel subgroup by 15.3.33.1. In particular ML acts on
Vz ~ E4, so we conclude n = 2. Then as AutM 1 (V2) ~ S3, L ~ L2(4).
Therefore L ~ L2(4) or L3(2) as case (2) of 15.3.37 holds, so Y2 = YL ~ A4. In
particular Y/0 2 (Y) is E 9 rather than 31+^2 as Y 2 has one noncentral 2-chief factor,
so Y = Y+ = Y 2 x Y.f ~ A4 x A4 fort ET - S, contrary to 15.3.9. D
Assume for the remainder of the proof of Theorem 15.3.35 that IE 1-t+,*· By
15.3.39, L :::! I, by 15.3.40, Y+ ::::; L, and by. 15.3.33.3, L 1:. M. Thus LS E 1-t+, so
I= LS by minimality of I. Let J+ := I/Or(L).
LEMMA 15.3.41. (1) F*(J+) = L+ is simple and described in 1.1.5.3.
(2) Mt is a 2-local of J+ containing a Sylow 2-subgroup s+ of J+ with YI ::::1
Mt, Y-ts+/02,w(Y-ts+) ~ S3xS3, andMt is maximal inf+ subjecttoF*(Mt) =
02(Mt).PROOF. Part (1) follows from 15.3.33.3. By 15.3.11.4 and Ooprime Action,
Mt = N 1 + (Y+). The remaining two assertions follow from 15.3.36.1 and Theorem
15.3.25. DLEMMA 15.3.42. L+ is of Lie type and cha~acteristic 2· ..
PROOF. Assume otherwise. If L+ ~ A1, then as Y+::::; Land Y+ is S-invariant,
Y+ ~ A4 x Z3. Thus Mr is the stabilizer in I of a partition of type 4, 3, as that
stabilizer is the unique maximal subgroup of I containing Y+S. This contradicts
F*(M1) = 02(M1) in 15.3.11.7.
By 15.3.11.8, O(L) = 1. Thus by the previous paragraph, L must appear incase (e) or (f) of 1.1.5.3. Inspecting the 2-local subgroups of the groups in those
cases for subgroups satisfying the conclusions of 15.3.41.2, we conclude that J+ ~
Aut(J2). Then as V::::; Z(R) by 15.3.36.2, and [V, Y+] is normal in Mr, we conclude
[V, Y+] ~ E4, and hence case (2) of Hypothesis 15.3.10 holds, so V 2 = [V, Y+l· But
now V1 ::::; 01(Y+) ::::; Or(L) from the structure of Aut(J2), so L::::; Oc(V1) ::::; M by
15.3.11.3, contrary to the choice of L. DWe are now in a position to complete the proof of Theorem 15.3.35.