10.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 753as It= 0
31
(Ca(u)) since I= 0
31
(H). Then CrTv(E) = C1•TJE), so that tacts
on C1TJE).Let P be the rank one parabolic of ITv over Tv not contained in M, and let
Pc and Pf be the rank one parabolics of L centralizing and not centralizing u,
respectively. Observe that as Lt = L 2 , t interchanges Y and Pc. If m(Vr) = 10,
then Cr·T; (u) is an L3(2) x L2(2) parabolic and CrTv (u) = (Y, P)Pc. Therefore as tinterchanges Y and Pc, and tacts on CrTv (E) = CrTv (u) by the previous paragraph,
P = pt. This is impossible, as (Y, P) is of type L 3 (2), while P Pc is of type
L2(2) xL2(2). Therefore m(Vr) = 5, and CrTv (u) = (Y, P, Pc) is of type L4(2); again
pt= P, and as Pf acts on 02 (P), so does Pj. This is impossible, as P centralizes
E, but PfP} contains a E 9 -subgroup D with CE(D) = 1 so m 3 (D0^2 (P)) = 3,
contradicting D0^2 (P) an SQTK-group. This concludes the treatment of the case
L 2:i: L3(2).
Therefore L '2:i: L 2 (4) and case (bl) or (b2) holds. In (bl), Vi = Vr ~ I,
so I ::; Na(Vi) ::; M by 10.2.6.2, contrary to I i. M. In (b2), [Vr, L] is the A5-module, so case (2) of 10.1.1 holds with Vi = [Vr, L]. Then Y := 0
31(CL 2 (v)) =/:-1,
and Y ::; I as 031 (H) = I by A.3.18. Hence 1 =/:-Y* ::; Nr• (L*) but Y* i. L*,
contradicting L * = 02 ( N 1 • ( L *). This contradiction finally completes the proof of
10.2.9. D
LEMMA 10.2.10. (1) For g E G - M, Vin Vf = 1.(2) IfCv(Lo) = 1, then Vi is a TI-set in G.
PROOF. As M permutes {Vi, V2} transitively and Vin Vi = Cv(Lo), (1) implies
(2).
Suppose g E G with 1 =/:-v E Vin Vf. By 10.2.9.1, Ca(v) ::; Mn M^9. Let
p be an odd prime divisor of 111, and for X ::; G let B(X) := OP' (x=). By
10.1.3, Lo = B(M), so LB ::; L 0 ; and Lo ::; Lg if v· E Cv 2 (Lo). In the latter case
g E Na(L 0 ) = M, so we may assume v rf. Cv 2 (Lo). Thus L = B(CL 0 (v)), so L^9 = L.
Then g EM by 10.2.6.2, establishing (1). DLEMMA 10.2.11. Assume case (3) of 10.1.1 holds with Cv(Lo) = 1. Let 1 =/:-
Vi E Cv; (Ti), set E := (vi, v2), and z := viv2. Let Gz := Ca(z), X := 02 (CL 0 (z)),
Kz := (X^0 z), and Vz := (E^0 •). Then
{1} Vz::; Z(02(Gz)) and Caz(Vz)::; NM(Vi).(2) If X < Kz then Vz E R2(Gz).
(3) V::; 02(Gz).
PROOF. By construction, z E Z(T), so Gz E 1-{e by 1.1.4.6. As XT ::; Gz,
02(Gz) ::; 02(XT) by A.1.6; then as 02(XT) ::; Ti ::; Caz(E), Vz ::; Z(02(Gz)).
Further
Caz(Vz)::; Caz(vi)::; NM(Vi)
by 10.2.10, since by hypothesis Cv(L 0 ) = 1, so (1) holds.
Set c; := Gz/Caz(Vz) and let R denote the preimage in T of 02(G;). By aFrattini Argument, Gz = Caz(Vz)Na(R). Thus if R::; Ti, then R centralizes E,
and hence also (ENaz(R)) = Vz, so that Vz E R2(Gz)· Thus to prove (2), we may
assume R i. Ti. In particular [X, R] i. 02 (X), so as Tis irreducible on X/02(X)and normalizes R, X = [X,R]. Thus X = [X,R] ::; R, so X* is a 2-group
and hence X = 02 (X) ::; Caz (Vz). By 10.1.3, X = 0
31