i3.9. CHAPTER APPENDIX: ELIMINATING THE A 10 -CONFIGURATION 97iorthogonal module for K/B ~ Ot(2) and ltKI = 6. Therefore t^0 nJ(M) = b.. 2 Ub.. 3
in view of 13.9.4. Thus all involutions in J(T) are fused tot or z. So writing I(8)
for the set of involutions in a subgroup 8 of G:
(!) t^0 n J(T) = I(Ti) U J(T2). So Ti and T 2 are the subgroups 8 of J(T)
maximal subject to the property that I(8) s:;; t^0. In particular each 4-subgroup of
J(T) consisting of members of t^0 contains either tor ts, and lies in either Ti or T 2.
Set Xi := 02 (CM(t)) = 02 (M2), X2 := 02 (CK(t)), X := 02 (Gt), and Gt :=Gt/(t). Thus Xi:::; X. We begin the explicit determination of X mentioned earlier.
Recall J(T) E 8yl 2 (Gt) by 13.9.2.6.
For i = 1, 2, Ai is the orthogonal module for Na(Ai)/Ai, with tNa(A,) the set of
nonsingular vectors in Ai, so Xi~ A4 with 02(Xi) = AnXi and t tJ: 02 (Xi)# s:;; t^0.
Thus we conclude from (!) that 02(Xi) :::; T2, so that 02 (Xi) =Ai n T 2 =Xi n T2,
and hence T2 = 02(Xi)02(X2):::; X.
If U is a 4-subgroup of Ti and g E Gt with U9:::; J(T), then t E U9, so U9:::; Ti
by(!). Hence I(Ti) is strongly closed in the Sylow group J(T) with respect to Gt, so
as Ti:::; Z(J(T)), N 0 t(Ti) controls fusion in J(T) by the Burnside Fusion Lemma
A.l.35. Then as Aut(Ti) is a 2-group and Ti is central in the Sylow group J(T),
each element of I'i is strongly closed in J(T) with respect to Gt; so by Thompson
Transfer, Tin X = 1 as X = 02 (X). Then (t)T2 E 8yl2((t)X), so by Thompson
Transfer, t tJ: X, and we conclude Tin X = 1. Hence T 2 E 8yb(X) as T2:::; X and
J(T) = TiT 2 is Sylow in Gt. Further Cx(tz) = Cx(z) = T n X = T2 ~ Ds, and
from the action of the Xi on the 02 (Xi), X has one class of involutions represented
by tz. Thus by I.4.1, X ~ L 3 (2) or A 6. In particular the involutions in X = 02 (Gt) ·
are in t^0.
As G is simple, by Thompson Transfer, s^0 n J(T) i=-0. We showed that z, t
are representatives for the G-classes of involutions in J(T), and thats tJ: z^0. Thus
s E t^0. We also saw that 02 (CM(s)) ~ A4, with J(0^2 (CM(s))) s:;; z^0. This is
impossible, as we saw J(0^2 (Gt)) s:;; t^0. This contradiction completes the proof of
13.9.5. D
Recall G := A 10 and set Q := 02 (0^2 (G2)). We may check directly from the
structure of A 10 that Q ~ Q~ and J(Q/(z)) = Q/(z) ~ E 16. Let Q := a-i(Q).
Since a : M ----+ M is an isomorphism:
LEMMA 13.9.6. Q ~ Q~ and Q = J(T) ~ E15.
Furthermore from the structure of A 10 , G2/Q ~ 83 x Z2. We wish to establish
analogous statements in G, starting with:
LEMMA 13.9.7. Gz,t = J(T).
PROOF. First by 13.9.2.6, J(T) E 8yl2(Gt)· As z E Z(T), F*(Gz) = 02(Gz)
by 1.1.4.6, so F*(Gt,z) = 02(Gt,z) by 1.1.3.2; then setting Gi,z := Gt,z/(t,z), we
obtain F(Gi , z) = 02 (Gi , z) from A.1.8. As the Sylow group J(T) of Gi is abelian,
J(T)*:::; Ca;)02(G;,z)):::; 02(G;,z),so J(T) = 02(Gt,z). Then the lemma follows from 13.9.2.2.