3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) S87We recall that M+T is a uniqueness subgroup in the language of chapter 1:
LEMMA 3.3.5. (1) M = Na(M+)·
(2) M = !M(M+T).(3) F*(M+T) = 02(M+T).
PROOF. Parts (1) and (2) are a consequence of 1.2.7.3 and 1.3.7. By definition
M+T E 1i(T), so (3) follows from 1.1.4.6. D
Throughout this section, we assume we are working in a counterexample toTheorem 3.3.1, so that Na(T) 1:. M. Our arguments typically derive a contradiction
by violating one of the consequences of 3.3.5.2 in the following lemma:
LEMMA 3.3.6. (a) D f:. M.
(b) 02((M+T,D)) = 1. Thus if 1 '/= X ~M+T, then D f:. Na(X).
(c) No nontrivial characteristic subgroup of Tis normal in M+T·
(d) Assume case (1) of Theorem 3.3.1 holds with L/0 2 ,z(L) of Lie type and
Lie rank 2 in characteristic 2. Then T acts on L unless possibly L/0 2 (L) ~ L 3 (2);
and if T acts on L, then (LT, T) is an MS-pair in the sense of Definition C.1.31.
PROOF. Part (a) holds as T:::; M, but TD= Na(T) 1:. M. Then (b) follows
from (a) and 3.3.5.2, and (c) follows from (b).
Assume the hypothesis of (d). Then unless L/0 2 (L) ~ L3(2), Tacts on L by1.2.1.3. Assume Tacts on L. Then (LT, T) satisfies hypothesis (MSl) in Definition
C.1.31 by 3.3.5.3, hypothesis (MS2) is satisfied as Tis Sylow in LT, and hypothesis
(MS3) holds by (c). DSet z := D1(Z(T)), v := (zM+) = (zM+T), M+T := M+T/CM+T(V), and
V := V/Cv(M+)·
LEMMA 3.3.7. (1) CM+T(V):::; 02,<P(M+T) and CT(V) = 02(M+T).
(2) J(T) 1:. CT(V), so V is a failure of factorization module for M+T.
(3) VE R2(M+T), so 02(M+T) = 1.
(4) [V,M+] = [Z,M+] and V = [V,M+]Cz(M+)·
PROOF. Since F*(M+T) = 02(M+T) by 3.3.5.3, part (3) is a conseque~ceof B.2.14. As V = (zM+), V = [V,M+]Z, so that V = [V,M+]Cz(M+) using
Gaschiitz's Theorem A.l.39. If M+ = 1, then V =Zand M+T:::; Ca(Z), contrary
to 3.3.6.c. Thus M+ '/= 1, so (1) follows from (3) and 1.4.1.5 with M+ in the role
of "Lo". If J(T):::; CT(V), then by B.2.3.3, J(T) = J(CT(V)) = J(02(M+T)) ~
M+T, contrary to 3.3.6.c. Thus J(T) 1:. CT(V), so V is an FF-module for M+T
by B.2.7. D
We now use 3.3.7 to determine a list of possibilities for M+ and V, which wewill eliminate during the remainder of the proof. Notice if case (2) of the hypothesis
of Theorem 3.3.1 holds, then conclusion (1) of the next lemma holds with Li~ Z3.LEMMA 3.3.8. One of the following holds:
(1) M+ = l;1 x L2 with Li ~ L2(2n), L3(2), or Z3, and Li = L2 for some
t E T-NT(L 1 ). Further [V,M+] = V16'V 2 , where Vi:= [V,Li], and either Vi is the
natural module for Li, the As-module for Li ~As, or the sum of two isomorphic
natural modules for Li~ L3(2).
(2) M+ ~ L 2 (2n) with n > 1, and [V, M+] is the natural module for M+.