594 3. DETERMINING THE CASES FOR LE .C.j(G, T)As D does not act on X, there is g E D - Na(X). Set G1 := XT, G2 .-
XBT, and G 0 := (G 1 , G 2 ). As XT/R 9,! S 3 and D acts on T, (Go, G1, G2) is a
Goldschmidt triple as in Definition F.6.1. Thus if g does not act on R = 02(XT),
02 (XT) # 02 (XBT), so Gt := G 0 /031(G 0 ) is described in Theorem F.6.18 by
F.6.11.2.
Suppose for each g ED -Na(X) that the group Gt defined by g satisfies case
(1) or (2) of F.6.18. Then 02 (G 0 ) =Rn RB is normalized by XT for all g ED, so
RD:= n Rd= n (Rn Rd)
dED dEDis invariant under XT and D, and hence RD::::; 02 (B) = T 0. Therefore as To::::; R,
RD = T 0 • Also 1>(T) ::::; Rn Rd since T/(R n Ra) 9,! Z 2 or E4 in cases (1) and
(2) of F.6.18, so 1>(T) ::::; To and hence 1>(T) = 1. Thus T acts on each Yi as
T nY.; # 1, sos= 1 and Y = F (B) is a simple SQTK-group by earlier remarks.
As 1>(T) = 1, we conclude from Theorem C (A.2.3) that Y 9,! L2(2n), J1, or L2(p)
for a prime p = ±3 mod 8. As XT / R* 9,! S 3 , the first two cases are eliminated.
In the third case B = Y as Y = F(B) and 1>(T) = 1. Thus XT 9,! Di2,
and NB(T) 9,! A 4. Then from the list of maximal subgroups of B in Dickson's
Theorem A.1.3, B = YT = (XT,XBT), contrary to our assumption that
each g ED - Na(X) defines a group Gt satisfying case (1) of (2) of F.6.18.
Therefore we may choose g E D - Na(X) so that Gt satisfies one of the
remaining cases (3)-(13) of F.6.18. In particular inspecting those cases, 1 # ct^00 =
E(Gt) is quasisimple. Then as 031(Go) is solvable by F.6.11.1, we conclude from
1.2.1.1 that Ko:= G 0 is the unique member of C(Go), and Kt= E(Gt). Hence
Ko E .C(G, T). By 1.2.4, Ko ::::; K E C(B), and K :::! B as T acts on K 0. As
T n Bo= 02(B 0 ), K # 1, so as Y is the unique minimal normal subgroup of B,
K = Y = F(B) is simple.
Assume for the moment that K;j < K. Set TK := T n K E Syl 2 (K). We
compare the possiblities for Kt described in F.6.18 to the embeddings described in
A.3.12, to obtain a list of possiblities for K*. Cases (2), (3), (15), (16), and (22) of
A.3.12 do not arise, since there the candidate "B/0 3 1(B)" for Kt does not appear
in F.6.18; this also eliminates the subcase of (8) with K* 9,! L 2 (p) for p = ±3
mod 8 and K 0 9,! A 5. In cases (4)-(7), (11)-(14), and (17)-(21), and also in the
remaining subcase of (8) where K 9,! L 2 (p^2 ), Aut(K) is a 2-group, so B = KT*
since K = Y = F(B). Furthermore in each case TJ< is self-normalizing in
Aut(K), so NB·(T) = T* in these cases.
Next assume we are in the subcase of (9) where K 9,! U 3 (5) and K 0 9,! A 6 • As
in the proof of 3.3.12, D induces a group of outer automorphisms of order 3 on
K centralizing TJ<, and as D normalizes T, T induces inner automorphisms on
K so that B = K D and TJ< = T. Now as D centralizes TJ< = T E Sylz(B*),
D centralizes 02(XT), so D normalizes the preimage Sin B of 02 (XT), and
hence as 02,z(K) is 2-closed, D normalizes 021 (S) = 02 (XT) = R, contrary to
the hypothesis of the lemma.
Finally in the remaining subcase of (9) and in (10), K* 9,! L3(p) with K 0 9,!
SL2(P) or SL2(P)/EP2 for an odd prime p, since K = Y is simple.
Thus we have shown that one of the following holds:
. (a) K 0 =K.
(b) NB·(T) = T*.