13.2. SOME PRELIMINARY RESULTS ON A 5 AND A 6 879PROOF. Part (1) follows from the structure of the A 5 -module. Then by (1),
R := 02(LT) E Syl2(0M(Vi)), so as O(G, R) :::; M = !M(LT), Na(R) :::; Mand
RE Syb(Oa(Vs)). Therefore by a Frattini Argument,Na(Vs) = Oa(Vs)(Na(R) n Na(Vs)),
so it remains to show that Oa(Vs) :::; M-since then Na(Vs) :::; Mv by 12.2.6. So
assume Oa(Vs) 1:. M. Then there is HE H*(T,M) with 02 (H) :::; Oa(Vs), and
hence R E Syl 2 (0^2 (H)R). Then by Theorem 3.1.1 there is 1 f= Ro :::; R with
Ro :::;! (LT,H), and so H:::; Na(R 0 ):::; M = !M(LT), contrary to assumption. DLEMMA 13.2.4. Assume n = 5. Then for any WE RdLT) with [W,L] f= l:
(1) Ri = (T n L)02(LT) = 02(0LT(Z n [W,L]). Further J(R1) = J(OT(W))
and Baum(R1) = Baum(OT(W)), so that O(G, Baum(R 1 )):::; M.(2) Let S := Baum(T); then either:
(a) S:::; OT(W) so thatJ(T) = J(OT(W)), O(G,S):::; M, andH*(T,M) ~
Oa(Z), or
(b) LT~ 85 , S = J(T) ~ E4 is generated by the two transvections inf',
(ZL) = V E9 Oz(L), and Ov(S) =Vi.PROOF. Recall that Hypothesis 12.2.3 excludes the groups in conclusions (2)
and (3) of Theorem 6.2.20. Thus case (1) of Theorem 6.2.20 holds, so for any
W E Rd LT) with [W, L] f= l, [W, L] is a sum of at most two A 5 -modules. Further
02(LT) is the kernel of the action of Lon both Wand V. Thus Ntr(Z n [W, L])is the Borel subgroup over f', so the first sentence in (1) holds. Next by B.3.2.4,
each member of P(T, V) is generated by transpositions, and hence none lie in R1.
Thus J(R1) :::; OT(W) = 02(LT), so that J(R1) = J(OT(W)) and Baum(R1) =Baum(OT(W)) by B.2.3.5; hence O(G, Baum(R1)) :::; M = !M(LT), so (1) holds.
Part (2) ·is essentially 5.1.2 applied to W in the role of "V". When J(T) :::;
OT(W), the final statement in (a) follows from Theorem 3.1.8.3. When J(T) 1:.
OT(W), the statements about Sand Vi follow from E.2.3. · D
LEMMA 13.2.5. If n = 5 then Na(Baum(T)):::; M.
PROOF. The lemma follows from 5.1.7. DLEMMA 13.2.6. If n = 5 then
(1) OT(v) E Syb(Oa(v)) for v E Vi - V1.
(2) Singular vectors of V are not fused in G to nonsingular vectors of V, so
that L controls fusion of involutions in V.
PROOF. Let v E V 2 -Vi. By 13.2.4.2, v E Vi:::; Ov(J(T)), so S := Baum(T):::;
Tv := OT(v); then S = Baum(Tv) by B.2.3.5. Let Tv :::; To E Syl2(0a(v)). Then
NT 0 (Tv):::; NT 0 (S):::; M by 13.2.5, so as Tv E Syb(Oivf(v)), Tv =To and hence (1)
holds. Then (1) implies that v ¢'. z^0 , where z is a singular vector in V, so that (2)
holds. D