3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN NG(T) 589(b) Y/02(Y) ~ L2(4), K/02(K) ~ Ji, D = (Kn D)Nv(Y), and ID :
Nn(Y)I = 7. Further T induces inner automorphisms on Y/0 2 (Y).(c) Y/02(Y) ~ A5, K/02(K) ~ U3(5), and D of order 3 induces an
outer automorphism on K/02(K) centralizing a subgroup isomorphic to the double
covering of 85 which is not GL 2 (5).
PROOF. Part (1) follows from 1.2.4 applied with Y, Hin the roles of "B, H".
By 3.3.6.b, Y < L. Applying 1.2.4 with Y, L in the roles of "B, H", and
comparing the embeddings described in A.3.12 to the list of possibilities for L in
3.3.8, we conclude that Y/0 2 (Y) is L 2 (2n), L 3 (2), A 6 , or L4(2). Furthermore Lisnot L 3 (2), so we conclude from 3.3.10.3 and 3.3.8 that M+ = L. Now by 1.2.8.1, T
normalizes Y, and then T also normalizes K. Thus (2) follows from 1.2.1.3.
Assume that conclusion (a) of (3) fails; we must show that conclusion (b) or
(c) of (3) holds. By (2), Y < K. Then Y/0 2 (Y) is described in the previous
paragraph, and the possible proper overgroups K of Y are described in A.3.12.
Set H* := H/Cs(K/0 2 (K)), and let Ys be the preimage of Y* in H. We
claim that Y :::;! Ns(Y*): By hypothesis, YE C(L, T), so Y is the unique member
of C(02(K)Y). Then as Y02(K) :::;! Ys, YE C(Ys) by A.3.3.2. Therefore as Tacts on Y, Y :::;! Ys by 1.2.1.3, establishing the claim.
By assumption, D i. NH (Y), so by the claim:
Nv·(Y*) = Nv(Y)*, so D* i. Ns•(Y*).
In particular, D* -=f. 1. Similarly Cv(K*) :S Cv(Y*) < D. Next TK := T n KESyl 2 (K) and 1-=f. D* :S Ns·(Tf<), so
Assume that K* is sporadic; that is, K appears in one of cases (11)-(20) of
A.3.12. Then Out(K) is a 2-group, so D :S K*, and we conclude from (**)
that K ~ Ji or J 2. In the latter case, Y ~ A5/2i+4 is uniquely determined by
A.3.12, and D :SY, contrary to(). In the former case, Ns·(Tf<) = Ns(T) is
a Frobenius group of order 21, and T* induces inner automorphisms on Y* ~ A5,
so that ID*: Nv·(Y*)I = 7. Thus D = (D n K)Nv(Y) and ID: Nv(Y)I = 7 by
(*). Then since the multiplier of Ji is trivial by I.1.3, K/02(K) ~Ji, so case (b)
of conclusion (3) holds. ·Thus we may assume K* satisfies one of cases (2), (4)-(9), (21), or (22) of
A.3.12. In cases ( 4)-(7), Out(K) is a 2-group, so that D :S K*, and (**) supplies
a contradiction. In case (2), K* is of Lie type and Lie rank 2 in characteristic 2,
with Y* = P*^00 for some T-invariant maximal parabolic P* of K*. Thus as there
are exactly two such parabolics,D :S 02 (Ns·(Tf<)) :S Ns·(P) :S Ns.(Y*),
again contrary to (*).In cases (21) and (22), Tf< is contained in a unique complement K]' to O(K*) in
K*, with K]' ~ S L 2 (p) for an odd prime p > 3. By the uniqueness of K]', Y* :S K]'
and D* acts on K]', so that Y* < K]' by (*). So replacing K by the C-componentKi of the preimage of K]', we reduce the treatment of these cases to the elimination
of the subcase of case (8) where H* ~ L 2 (p) for some prime p = ±3 mod 8 and