1060 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTYAssume (3) fails, and adopt the notation of section B.3 to describe UH. Now
. L 1 T induces L 2 (2) on V ~ E 4 , so as we saw in the proof of 14.6.10, either
(i) Lj_T* is the stabilizer in H* of the partition A:= { {1, 2}, {3, 4}, {5, 6}, {7} },
V2 = (e1234), and V= (e1234,e1255), or
(ii) .Lr1'* is the stabm~e~ 'of tli~ ~artition e := { {1, 2}, {3, 4}, {5, 6, 1} }, -cs =
(e5,5), and V = (e5,6, e5,7)·However in case (i), m 3 (CH(Vi)) = 2, contrary to 14.7.4.3, so case (ii) must hold.
Here H 0 ~ A 5 stabilizes {5, 6}, and UH= (VH^0 ), contrary to (1). D
LEMMA 14.7.38. U"Y > D"Y.
PROOF. Assume u"Y = DT By 14.5.18.1, UH induces a nontrivial group of
transvections on U"Y with center Vi. Recall that bis odd by 14.7.3.1, so by edge-
transitivity in F.7.3.2, we may pick g = 9b E (LT,H) such that g: bb-VY) 1--+
('Yo, '°Y1). Let /3 := '}' 1 g, so that Uf3 induces a group of transvections with center
B1 := V{ on UH. By (1) and (2) ofF.9.13, Uf3 :::; 02(G'7 0 m) = Ri. Set Hi := (U/f).
If Hi is solvable then by G.6.4, Hi is a product of copies of 83, so by 14.7.28,
Li = 02 (Hi) and hence Hi = LiUJ ~ 83 , contradicting Uf3 :::; R 1. Therefore
Hi is not. solvable. Thus by 1.2.1.1, K* = [K*, UJ] for some K E C(H). LetUK:= [UH,K]. As UJ induces transvections on UH, G.6.4 says UK/CuK(K) is a
natural module for K*UJ/CK*U~(UK) ~Sn or Ln(2).Suppose first that K* ~ A 5 or L3(2), and let L'K be the projection of Li in
K with respect to the decomposition K x CH(K). As L 1 is T-invariant, L'K is
T-invariant; so either L'K ~ A 4 , or L'K = 1 so that [Li, K] = 1. In case K* ~ A5,
as UJ induces a transposition on K and Uf3 :::; R 1 , L'K = 1, so [Li, K] = 1. In case
K ~ L3(2), as Li is T-invariant and L'K = [L'K, T nK], either Li = L'K :::; K or
[Li, K*] = 1. However if [Li, K*] = 1, then [UK, Li]= 1 since EndK·(UK) ~ F2,
so V = [if,L 1 ]:::; CuH(K), and then UH= (VH):::; CuH(K), contradicting K*-/:-1.
Therefore Li :::; K* ~ L3 (2). Further V = [if, L 1 ] :::; fj K, so that fj K =UH. Then as EndK•(UH) ~ F 2 , CH(K) = 1 as H* is faithful on UH, so that
H = KT*. Then as the natural module UH is T-invariant, we conclude that
H* ~ L3(2), contrary to 14.7.11.
Therefore K*UJ ~ 85, 81, Ss, L4(2), or L5(2). In particular by A.3.18, K =
031 (H), so L 1 :::; K, and then as above, UK= UH and H = KUf3*. By 14.7.11,
H is not 85, and by 14.7.37, H is not Ln(2) or 87.
Thus it remains to eliminate the case H* ~ 88 • Here V projects on a singular
line in the orthogonal space UH/CuH(H), so V 2 projects on a singular point; henceCH*(if2)/02(CH*(i12)) ~ 83 wr Z2,
contrary to 14.7.4.3. DIn view of 14.7.38, we establish the following convention:
In the remainder of the section, we adopt Notation 14.1.1.
REMARK 14.7.39. Whenever we can show that m(U;) = m(UH/DH), our
hypotheses are symmetric in '°Y and '°Yli see Remarks 14.7.17 and F.9.17 for a more
extended discussion of this point.