8i8 i2. LARGER GROUPS OVER F2 IN £.'f (G, T)
He, or Ru. To rule out £: 2 (49) or (8)L3(7), observe that in those groups, some
element of T induces an outer automorphism on Li/0 2 (Li) ~ L3(2), while as
f' :::; L, this is not the case in LT. In the remaining cases by A.3.18, Ki is the
characteristic subgroup ()( Gi) of Gi generated by all elements of order 3. Hence
Ki contains I := 031 (Mi), since I/02(I) ~ Z3 x L 3 (2) by 12.5.2.1. Further
Mi = IT and Mi/0 2 (Mi) ~ 83 x L3(2), whereas Aut(Ki/02(Ki)) contains no
such overgroup of a Sylow 2-group. This completes the proof of the lemma when
i = 1, although we need some more information about Ki which we develop in the
next paragraph.
Set (KiT) := KiT/02(KiT). When Ki ~ M24, L5(2), or J4, the over-
groups of T are described by a 2-local diagram, cf. [RSSO] or [Asc86b]; we now
describe the embedding of LiT and Mi in Ki in terms of the minimal parabol-
ics in the sense of Definition B.6.1 indexed by the nodes of of this diagram: As
LiT /02(LiT) ~ L3(2) and Mi /02(MJ.) ~ L2(2) x L3(2), it follows that if
Ki ~ L5(2) then (up to a symmetry of the diagram) LiT is generated by the
third and fourth minimal parabolics of Ki, and the remaining parabolic P of
Mi is the first parabolic of Ki. If Ki ~ M24 or J4, then Lj_T is generated by
the parabolics indexed by the "square node" and by the adjacent node in those
diagrams. Further in M24, P* is the parabolic PK: indexed by the node whi~h is
adjacent to neither of these nodes, while in J 4 , the corresponding parabolic PK sat-isfies PK:/02(PK:) ~ 85, and P* is the Borel subgroup of that parabolic. Thus when
Ki ~ M24, Mi is the trio stabilizer in the language used in chapter H of Volume
I, while when Ki~ J4, PK:Mi ~ 85 x L 3 (2)/2^3 +1^2 and Mi~ (84 x L 3 (2))/2^3 +i^2.We next treat the cases i = 3 or 4. Here Li/CLJV;,) = GL(Vi) by 12.5.2,
so that Ki = LiCKi (Vi). Hence if Ki/02(Ki) is quasisimple, then Li = Ki, as
required. Therefore we may assume that Ki/0 2 (Ki) is not quasisimple, and it
remains to derive a contradiction. As Kif 02 (Ki) is not quasisimple and Li :::; Ki,
we conclude from A.3.12 that i = 3, K3/02(K3) ~ 8L 2 (7)/E49, and K 3 = XL3
for X := B1(K3). Set Y := 02 (P). Recall X char K3 <J G3. and 02(X) -=f. 1 using
1.1.3.1. Also X centralizes Va :2: Vi, so X :S Ki,3 := 02 (CK 3 (Vi)) :S Gin G3.
We saw earlier that Y = 02 (P) :S L 3 , so Y :S Ki,3· Then Ki, 3 T/02(Ki,3T) ~
GL2(3)/E49 and Ki,3 = YX = (YK^1 ,^3 ).
Suppose first that Ki is L5(2), M24, or J4. We saw earlier that Y:::; Ki. HenceKi,3 = (YK^1 •^3 ) :S Ki, so Kl.' 3 T is a subgroup of KiT containing T*. But from
the description of overgroup~ of a Sylow 2-group in terms of the 2-local diagramsfor L5(2), M24, and J4 mentioned earlier, no such group has a GL 2 (3)/E 49 -section.
So we may suppose instead that Ki= Li or Ki~ SL 2 (7)/E 49. By an earlier
observation [Li, Y] :S 02(Li). Thus if Ki =Li, Y centralizes Ki, and we claim this
also holds when Ki ~ SL2(7)/E49: For Ki = B1(Ki)Li and there is Y <Yo :::; Pwith Yo/02(Y) ~ L2(2) and [L1, Yo] :S 02(L1). Then as Li ~ 8L2(7) is centralized
in Aut(B1(K1)*) by Z(GL 2 (7)) ~ Z 6 which is abelian, we conclude Y = [Yo, Yo]
centralizes 37(Ki)· Hence Y = 07 (Y) centralizes Ki= 071 (Ki), establishing the
claim.
By the claim, (YK^1 •^3 ) = Ki,3 centralizes Ki, so as X:::; Ki,3, X centralizes
Ki- By construction X E B(G, T), so by 1.3.4, either X :::;! Gi, or X < Ko E
C(Gi) with m3(Ko) = 2. In the latter case Ko = (XK^0 ) centralizes Ki/0 2 (Ki),