814 12. LARGER GROUPS OVER F2 IN .Cj(G, T)LEMMA 12.4.6. If L ~ L3(2), then Rl E Syb(Gv) and IT: Rll = 2.
PROOF. First R 1 E Syl 2 (Mv) and IT : Rll = 2. Thus the result holds if
Na(R 1 ) :::; M. So we assume that Na(R 1 ) i. M, and it remains to derive a
contradiction.Let Z1 := f21(Z(R1)). Then by 12.4.5.2,
Z1 = Czq (R1) = ZCv(R1) = Z(v).
Now [Z, L] = 1by12.4.5.1, so Zn Z9 = 1 for g E Na(R1) - M by 1.2.7.4. Henceas Z is a hyperplane in Z 1 , we conclude Z is of order 2, so that Z = Zv and
E = Z1 = f21(Z(R 1 )). In particular, ·
Na(R1) :::; Na(E).Furthermore Ca(E) :::; Ca(Z) :::; Gz = M, and Auta(E) ~ 83 with AutM(E) of
order 2; thus INa(R1): NM(R1)I = 3.
Next as Na(R 1 ) i. M = !M(LT), there is no nontrivial characteristic subgroup
of R 1 normal in LT. Thus (LR 1 , R 1 ) is an MS-pair as in Definition C.l.31, so that
C.1.34 applies. As Vis indecomposable, conclusion (5) of C.1.34 holds; hence L is
a block with Q = VCr(L) and CR 1 (L1) = ECr(L).
Suppose that Na(R1) :::; Na(L 1 ). Then Na(R1) normalizes (CR 1 (L1)) =
(ECr(L)) = (Cr(L)), so (Cr(L)) = 1 since Na(R1) i. M = !M(LT). Thus
Cr(L) is central in VCr(L) = Q and in Q(TnL) = T. We conclude Cr(L) = Z =
Zv, so that Q = VCr(L) = V. Since the nontrivial characteristic subgroup J(R1)
of Rl is not normal in LT, J(R 1 ) i. 02 (LT) = Cr(V), so there is A E A(R1) with
JI =f. 1. By 12.4.5.3, JI = R 1. Thus A(R 1 ) = {A, V} by B.2.21, since V is self-
centralizing in G and Cv(A) = Cv(a) for a E JI# by B.4.8.2. Hence 02 (Na(R1))
acts on V, so 02 (Na(R1)):::; M, contradicting /Na(R 1 ): NM(R 1 )1=3.
Thus there is g E Na(R1) - Na(L1). We have seen that Na(R1) :::; Na(E)
and Ca(E) :::; M; so as Ll :::;! CM(E) while m3(CM(E)) :::; 2, LlLf =: X =
e(Ca(E)) and X/0 2 (X) ~ E 9. Then Xo := Cx(L) is of order 3, so by C.1.10,
X1 := 02 (Xo) centralizes L and Xi/0 2 (X 1 ) ~ Z3. Next X1 is centralized by
t E TnL-R1 inverting Li/02(L1), so L 1 and X1 are the two T-invariant members
of the set Y of subgroups Y of X such that Y = 02 (Y) and IY: 02 (Y)I = 3. Now
Na(R1) normalizes X and hence permutes Y. Since Nc(R1) i. Na(L1), while Li
is stabilized by NM(R1) of index 3 in Na(R1), the Na(R1)-orbit of Ll has length
3, and the fourth member of Y is fixed by Na(R 1 ). Since T :::; Na(R 1 ) and X 1
is the only T-invariant member of Yother than X1, we conclude X 1 :::;! Na(R 1 ).
However X 1 :::;! XLT, so
Na(R1):::; Na(X1):::; M = !M(LT),
contrary to our earlier reduction. This completes the proof of 12.4.6. D
LEMMA 12.4.7. L controls fusion of involutions in V.
PROOF. Suppose first that L ~ L 3 (2). By 12.4.6, vG n Zv = 0. Thus as Lis
transitive on V - Zv, the lemma holds in this case.
Next take L ~ G 2 (2)'. Then Zn Vis a 4-group containing a representative
of each of the three orbits of Mon V#. But Na(T) :::; M by Theorem 3.3.1, and
Na(T) controls fusion in Z by Burnside's Fusion Lemma A.1.35, so the lemma
holds in this case also. D