980 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTYPROOF. SupposeM 1 EM(0 2 ,F•(M)T)andletH:=MnM1. As02,F*(M) _::::;
H, 02(M) = 0 2 (H) by A.4.4.1. Thus M =Mi by 14.1.14. D
LEMMA 14.1.16. IfT _::::; H _::::; M with 1=/:-0 2 (H) and C(M,02(H)) = H, then
M= !M(H).
PROOF. Suppose M 1 E M(H) - {M}. Then JM(H)J > 1, so 02,F•(M) _::::; H
by 14.1.13, contrary to 14.1.15. D
LEMMA 14.1.17. Let M 1 E M(T) - {M} and assume either
(a) M 1 :SM and V = V(M), or
(b) Mi= Mc.
Let R := 02 (M 1 n M), assume there is T-invariant subgroup Yo of M with Yo ofodd order, and set Y := 02 ((RY^0 T)) and M* := M/02(M). Then
(1) fl =f:. I.
(2) Y = [Yo,fl].
(3) [CM(V), Y R] = 1 and Cy(V)::::; O(Z(CM(V) R)).
(4) If 1 =/:-r ER is faithful on Op(M) for some odd prime p, then CM (r*)
has cyclic Sylow p-groups, so mp(CM(V)) ::::; 1.(5) R = 02(CM(V)R), so Nr;; 1 (R) = NM(R).
PROOF. In case (a), M 1 ;SM and V = V(M), so CM(V) _::::;Mi by A.5.3.3. In
case (b), Mi =Mc and CM(V) ::::; CM(Z n V) ::::; Mc= !M(Cc(Z)). So in either
case, CM(V) ::::; Mn M 1 ::::; NM(R). Since V E R 2 (M) by 14.1.1, it follows thatCR(V) = 02(CM(V)) = 02(M). Then R = 02(CM(V)R), and hence (5) holds.
Further if R = 1, then R = CR(V) = 0 2 (M), contrary to 14.1.14. Hence (1) is
established.Next by Coprime Action, Y 0 = Y+Y, where Y+ := Cy- 0 (.R) and Y := [Yo,R]
are T-invariant since Yo and R are T-invariant. By (5), Y+ ::::; NM(R), so YR :=
(.RYoT) = (flY-) and YR = fly with Y = Y. In particular (2) holds.
Also [CM(V), R] ::::; CR(V) = 02(M), so [CM(V), R] = 1, and hence Y* =
[Y, R] centralizes CM(V)*, so that (3) holds. Part ( 4) follows from A.1.31.1
applied to the product of a Sylow p-subgroup of Op(M*)CM* (r*) with (r*). DLEMMA 14.1.18. Let M :=Mt as in 14.1.12, and assume V := V(M) is of
, rank 2 with M ~ 83. Let Re := 02 (M n Mc), Y := 02 ( (R~) ), R := CT(V), and
M* := M/0 2 (M). Then
,(1) Mis the unique maximal member of M(T) under :S.
(2) RcR = T and Mn Mc = CM(V)Rc·
(3) Y = 02 (M) ~ Z 3 and 02(Y) = Cy(V).
(4) M = !M(YT).
(5) M = Y R~ x CM(V) with Y R~ ~ 83 and m3(CM(V))::::; 1.
(6) Z is of order 2 and Mc= Cc(Z).
(7) M(T) = {M,Mc}·
PROOF. As V = V(M) is of rank 2 and M ~ 83 , Z is of order 2, so (1)
follows from part (1) of 14.1.12. Further Mc =/:-M by part (4) of that result, so
case (b) of the hypothesis of 14.1.17 holds. By 14.1.17.1, Re =j:. 1, so as f' is of
order 2, Re= f' and hence T = RRc· As CM(V) ::::; Cc(Z) ::::; Mc= !M(Cc(Z)),