1140 15. THE CASE Cf(G, T) = 0of 15.3.41, using the same proof, but replacing the appeal to 15.3.36.1 by an appeal
to (a):
(e) F*(J+) = L+ is simple, and is described in 1.1.5.3. Further Mt is a 2-local
of J+ containing a Sylow 2-subgroup 3+ of J+ with Yt = 031 (Mt), 3+y: jR+ ~
8~, and Mt is maximal subject to F*(Mt) = 02(Mt).
We now eliminate the various possibilities for L+ arising in 1.1.5.3 and satisfying
condition ( e).
Suppose first that L+ is of Lie type over F 2 ,,,, and hence is described in cases
(a)-(c) of 1.1.5.3. Then Mt is a maximal 8-invariant parabolic by (e).
Assume that n > 1. Then as Y+ = 031 (ML), we conclude L+ ~ L 3 (2n)
or L 2 (2n) with n even, or U 3 (2n), ·and Mt is a Borel subgroup of L+. Then
E4 ~ Vi = [V, Y+] :::'.) ML by (b), so we conclude that n = 2. But from the
structure of Aut(L), Cs(Y+) = Cs(L), contrary to (c).
Son= 1. As Y+ = 031 (ML) and ML is a maximal 8-invariant parabolic, either
L+ is of Lie rank 2, or J+ ~ 88 and Mt is the middle-node minimal parabolicisomorphic to 83 /Q~. As E 4 ~Vi :::'.) M 1 , the last case is eliminated. Now by (b),
Vi :::; Cs(Y+), but [V 1 , L] -=f. 1 by (c), and again Vi :::'.) M1, so we conclude that
J+ ~ 86. However in this case Y+ ~ A4, contrary to (d).Suppose next that L+ is sporadic. We inspect the list of possible sporadics
in case (f) of 1.1.5.3 for subgroups J+ of Aut(L) such that there is a 2-local Mt
satisfying (e) and with E4 ~Vi =[Vi, Y+] :::'.) M1. We conclude L+ ~ Mi2. But
then Cs(Y+) = Cs(L), contrary to (c).
If L+ ~ A 7 , then arguing as in the sporadic case, M1 is the stabilizer of a
partition of type 23 , 1, so Y+ ~ A 4 , again contrary to (d).
From the list of 1.1.5.3, this leaves the case where Lis L3(3) or L2(p), pa Fermator Mersenne prime; we may take p > 7 as L 2 (5) ~ L2(4) and L 2 (7) ~ L 3 (2) were
eliminated earlier. However in each case, there is no candidate for M1 satisfying( e). This completes the proof of 15.3.43. D
By Theorems 15.3.35 and 15.3.43:
THEOREM 15.3.44. Assume Hypothesis 15.3.10. Then M = !M(Y+8).In the remainder of this subsection we deduce information about the structureof M and of members of H(T, M) from these uniqueness results.
THEOREM 15.3.45. For i = 1, 2:{1) M = !M(Cy(Vi)8).
(2) Na(Vi) :::; M.
{3) Ca(Cv,(8)):::; M.
PROOF. First Hypothesis 15.3.10.1 is satisfied with Y+ := Y, so by 15.3.44,
M = !M(YS). Therefore Hypothesis 15.3.10.2 also holds with Y+ := 02 (Cy(Vi_)),
so (1) holds since V 1 and Vi are conjugate in M. Then as Vi :::'.) Y8 and Cy(Vi)8:::;
Ca(Cv,(8)), (2) and (3) follow from (1). DRecall that we view V as a 4-dimensional orthogonal space of sign + 1 over
F2, and M as the isometry group of this space. In particular, there are two M-classes of involutions in V: the 9 singular involutions fused to z under M, and the
6 nonsingular involutions in vt U ~#. We will show next that these classes are not