15.3. THE ELIMINATION OF Mf/CMr(V(Mr)) =Sa wr Z 2 1137Nc(Vi) = I, so as M1 is transitive on V 1 #, V 1 is a TI-subgroup of G by I.6.1.1.
Then as V1 ~ E4:(*) Vi is faithful on any subgroup F = 02 (F) on which it acts nontrivially.
Recall T is fused to an element of W 0. But L is transitive on the involutions
in f L, and each involution in L is fused into V under L and hence is in z^0 U v^0.
Thus to obtain a contradiction and complete the proof, it remains to show that T
is not fused to f, v, or z.
Recall that T f centralizes Vi. Suppose T f = v9 for some g E G. Then Vi
normalizes V{ sin~e V 1 is a TI-subgroup of G, and hence by I.6.2.1, [Vi, V{] = 1.
Thus V{ :S Ca(Vi) = Cr(Vi) = L(Tf), so V{ centralizes F := 02 (CL(Tj)) ~ E 9
by(*). Then as ICAut(L)(F)I = 2, 1 #-V{ n Vi, so V{ =Vi as Vi is a TI-subgroup,
contrary to v^9 = T f tj. Vi. Thus T f is not fused to v, so as Tt = T f, T is not fused
to v. Similarly using (*) and Generation by Centralizers of Hyperplanes A.1.17,
O(Cc(Tf)) centralizes V1, so O(Cc(Tf)) :S O(Cr(Tj)) ~ Eg.
Next 02 (C£(T)) ~ L 2 (4), so applying I.3.2 as above, we conclude F*(Cc(T)) #
02(Ca(T)), and hence Tis not fused to z in G. Therefore Tis fused to fin G, so
Tf is also.
Let Lt := 02 (CL(f)) and Gt := Cc(!); then Lt ~ L3(2), and again using
I.3.2, Lt :::; 021,E(Gt)· We saw earlier that O(Cc(Tf)) is an elementary abelian
3-group of rank at most 2, and f is fused to Tf; so 021,E(Gt)^00 = E(Gt), and
hence Lt :::; E(Gt ).
Suppose that there is a component Li of Gt of 3-rank 1. As f is fused to T f,
and a Sylow 3-subgroup of C 1 (Tf) is isomorphic to 31+^2 , there is 31+^2 ~ B:::; Gt·
But now m3(L1B) = 3 from the structure of Aut(L 1 ) with Li of 3-rank 1 in
Theorem C (A.2.3), contrary to Gt an SQTK-group. Thus no such component
exists. But Lt:::; E(Gt) so there is a component L 2 of Gt which is not a 3^1 -group,
and then as m 3 (Gt) :::; 2, L 2 is of 3-rank 2 and L2 = 0
31
(E(Gt )), so Lt :::; L2.
Indeed L3(2) ~ Lt is a component of CL 2 (v), so we conclude using Theorem C
that L2/Z(L2) ~ L3(4), J2, or L3(7). By A.3.18, either Cc 1 (L2) is a 3'-group or
L 2 /0 2 (L2) is SL3(4) or SL 3 (7), with O(Z(L2)) the unique subgroup of order 3 in
Cc 1 (L2).
As T f is fused to f, there is a conjugate U1 of Vi with L3 := 02 (Cc( (U1, !) ) ) =
O(Cc 1 (U 1 )) ~ 31+^2. In particular, U1 acts nontrivially on L2, and hence U1 acts
faithfully on L2 by (*). By the previous paragraph, either L3 is faithful on L2
or Z(L 2 ) = Z(L 3 ). We conclude from the structure of centralizers of involutions
in Aut(L 2 ) in our three cases that L3 = O(Cc 1 (U1)):::; Cc 1 (L2), contrary to the
previous paragraph. D
LEMMA 15.3.39. L =Lo.
PROOF. Assume L < L 0. Then Lo := LL^8 for some s E S - Ns(L), with
L described in 1.2.1.3. Then m 3 (L) = 1 by 15.3.34, and by 15.3.33.3, L is also
described in 1.1.5.3 with O(L) = 1, so Lis simple and Lo= L x L8. Therefore as
m 3 (Y+) = 2 and Y+ :::; Lo by 15.3.37, Y+ = xxs where X := 02 (Y+ n L). Next
there is Y2 :::; Y+ n M2, with 112 = [V, Y2] and Y 2 is S-invariant. As Y2 and 112 are
s-invariant, they are diagonally embedded in L 0 , so X is the projection of Y2 on
L, and VL = [VL, X], where VL is the projection of 112 on L. Similarly s acts on