1136 i5. THE CASE .Cf(G, T) = 0L 3 (2n) with n even. As SE Syb(I) acts on Y+, Y+S acts on the Borel subgroup
B of L over Sn L = SL, and then YL := 02 (Y+ n L) ::::; B ::::; M 1 by 15.3.33.1.
Hence by 15.3.11.6, M1 = N1(B), n is coprime to 3, and YL/02(YL) ~ Z3. Also
[V, YL] I-1: for otherwise V::::; Cs(YL) = Cs(L), so L:::; Na(V) = M, contrary tothe choice of L. Since YL is S-invariant with YL/0 2 (YL) of order 3, YL :::; Mi for
i = 1or2, and we may choose notation so that i = 2. Thus E4 ~ Vz = [V, YL] :::l B,
son= 2, and v 2 + is the root group Z(St) of£+. Further Vi= Cv(YL):::; Ca(L),
so Na(Vi) i. M, and hence by 15.3.11.3, case (1) of Hypothesis 15.3.10 holds, soY = Y+ and YL = 02 (Y n M2)· Hence Y/02(Y) ~ Eg. Let YD := 02 (Y n Mi)
and t E T - S. Then Yi = 02 (Y n M 2 )t = 02 (Y n Mi) = YD, and Y = YLYD.
As case (3) of 15.3.37 holds, an element of order 3 in YD induces a diagonal outerautomorphism on£+. Also YD:::; Mi:::; Ca(Vz), so from the structure of PGL3(4),
V/ = Cs+(YD) L and SL= [SL, YD]. Of course [SL, YD]:::; 02(YD), so as 02(YL) =
SL and Yi =YD, SL = 02(YD) by an order argument. Thus tacts on SL, andhence also on Z(St) = v;+. Since t interchanges Vi and Vz, Vi :::; VzZ(L).
NowCz(L)(YD) = 1, and we showed v2+ = Cs+(YD)· L Further Z(L) = CsL(YL)·
Thus V2 = CsL(YD) and Z(L)t = CsL(Yl) = CsL(YD) = Vz ~ E4, so E4 ~Vi=
Vi= Z(L).
Next Cs(YL) = Cs(L), so conjugating by t, ICs(YD)I = ICs(L)I, and as Y =
YLYD, Cs(Y) = Cs(L) n Cs(YD)· Then as Cs(YD) :::; VzCs(L) and IVzCs(L) :
Cs(L)I = IVzl = 4, IVzCs(L): Cs(YD)I = 4. Therefore ICs(L): Cs(Y)I = ICs(L):
Cs(L) n Cs(YD)I :S 4, so as Cv 1 (YD) = 1, Cs(L) = ViCs(Y). But by 15.3.8,
f"h(Z(S)) =Zs:::; V, and Cv(Y) = 1, so we conclude that Cs(Y) = 1; hence Vi=Cs(L). Thus CT(V) = 02(YS) = 02(Y) =SL. As YS/CT(V) = YS ~ S3 x S3,
we conclude that £+y_t s+ = Aut(L+). As O(J) = 1, Vi = Cs(L) = C1(L), so
L = F*(I) and hence I= LYDS.
Let X E Syl3(Y); then CsL (X) = 1, and by a Frattini Argument, Nys(X) ~
S3 x S3. Let E := Ns(X); then E = (T, !), where T and fare involutions inducing
a graph and a field automorphism on£+, respectively. Further X =XL x XD,
where XA := X n YA for A := L,D, f inverts X, and Tf centralizes Vi. By a
Frattini Argument, we may choose t E NT(X), so (t, T) ~ f' ~ D 8. Thus we may
choose t to be an involution and Tt = T f.Let w be of order 4 in (T, t), and set W := (w, SL); then IT : WI = 2, so as
G = 02 (G), T^0 nw I- 0 by Thompson Transfer. As W/SL = W ~ Z 4 and w^2 = f,
Tis fused to a member of Wo := (f)SL.
Next Vi= Z(L) :::l I. Then Ji := Na(Vi) E 1-f.+, so SE Syl2(li) by 15.3.11.1,
and thus L :::; Li E C(Ii) by 1.2.4. Then m 3 (Li) 2 m 3 (L) = 2, so applying our
reductions so far to Ii, L1 in the roles of "I, L", we conclude that Ii/0 2 (Ji) ~
Aut(L3(4)), and hence L = L1 and I= Ji. That is, I= Na(Vi).
Let (v) = Zs n V1, Gv := Ca(v), and Gv := Gv/(v). Then Z2 ~ V1 and
LS= C 0 .., (V1) as I= Na(V1). By I.3.2, L :S 021,E(Gv), so F*(Gv) I-02(Gv), and
hence IGvl2 < ITI as G is of even characteristic. Thus as S:::; Gv and IT: SI = 2,
SE Syl2(Gv)· Therefore L::::; Lv E C(Gv) by 1.2.4, with the embedding described
in A.3.12. As L :::; 021,E(Gv), Lv is quasisimple. Indeed as Lis a component of
Ct.., (V1), while the only embedding of L 3 ( 4) appearing in A.3.12 is in M23, and M23
has trivial Schur multiplier by I.1.3, we conclude Lv = L. Thus Gv :::; Na(L) :::;