i046 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTYPROOF. Assume otherwise and let Hi := LiT and Hz the minimal parabolic
of H over T distinct from H 1. Set Y := oz(Hz). Then H = (L1T, Y) i. M, so
Yi. M. Thus [V2, Y] = 1by14.5.3.2, so YLz/Oz(YLz) ~ Eg by 14.7.4.2.
Let Do := H, Di := HzLz, Dz :=LT, F := (Do, D1, Dz), and D := (F). We
will show (D, F) is an A 3 -system or 03 -system in the sense of section I.5.
Set Pi := Oz(Di) and Di := Di/Pi. We saw oz(iJ1) = Lz x Y ~ Eg.Further Oz(Gi) ~Po by A.1.6, so Lz = [Lz,Po] by 14.7.4.2. On the other hand,
Y ~ H ~ Na(Po), so Po centralizes Y, and hence Po is of order 2. Next from the
structure of H under our hypothesis, Y = [Y, T], so Y = [Y, T], and hence
Di~ Lz(2) x Lz(2). Of course Dz/Qz = LT/Oz(LT) ~ L3(2), so hypothesis (D2)
of section I.5 holds. By the hypotheses of this lemma, hypothesis (Dl) holds, and
by construction hypothesis (D3) holds. By definition, D = (F). As H i. M =
!M(LT), kerT(D) = 1, so hypothesis (D4) is satisfied. As V1 ~ Z(H), hypothesis
(D5) holds. This completes the verification that (D, F) is an A 3 -system or 03-
system.As (D, F) is an A3-system or 03 -system, D ~ L 4 (2) or Sp 6 (2) by Theorem
I.5.1. But then Oz(H) is abelian, contrary to 14.7.4.1 as Oz(Gi) ~ QH. D
Recall that when D'Y < U-p we adopt Notation 14.7.1, and in par_ticular we
obtain a with Ua ~ Ri.
LEMMA 14.7.12. (1) L acts 2-transitively on the subgroups Uff.-generating S.
(2) Assume D'Y < U'Y and b = 3. Then Uff.-={UH} U U[;_^1 T.
PROOF. As NL(UH) = H n Lis a maximal parabolic of Land L/Oz(L) ~
L3(2), Lis 2-transitive on L/NL(UH), so that (1) holds.
Assume the hypotheses of (2). As b = 3, 'Y E I'('Yz), so a = "fh E I'('Yzh) =
I'('Yo), and hence Ua E Uff.-. Therefore (2) follows from (1). DLEMMA 14.7.13. (1) Set E := [UH,Q] and R :=(EL). Then [S,Q] = R.
(2) Assume [E, Q] = V. Then [R, Q] = V.
Assume further that D'Y < U'Y and b = 3.
(3) If [E, Ua] = 1 then R ~ Z(S).
(4) Set A := [UH, Ua] and B := (AL). Then 4'.i(S) = [S, SJ = B.
PROOF. Observe that (1) and (2) follow directly from the definitions of S =
(UfJ.-) and R = \EL). Now assume that D'Y < U'Y and b = 3, so that in particular
Notation 14.7.1 holds. Suppose E commutes with Ua. As E also commutes withUH and E :'::l LiT, E ~ Z(S) by 14.7.12.2. Thus (3) holds. Similarly 14.7.12.l
implies (4). DWe close Subsection 1 with a brief overview of an argument used to analyze
the most difficult configurations in Subsections 2 and 5:
(a) Begin with a particular structure for H*, and possibly i) H.
(b) Determine the structure of QH, and hence of H-cf. 14.7.20 and 14.7.71.1.
(c) Determine the structure of S, and hence of LT-cf. 14.7.24, 14.7.25, and
14.7.71-14.7.72.