1002 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN £,f(G, T) IS EMPTYfrom section 12.8 with Vi in the role of "Vi". Similarly V1 Vf can play the role of
"V2"·
LEMMA 14.3.28. (1) Z(Ib) = Z[j n Zf;..
(2) Zu n Z(Ib) = 1.PROOF. As (U, U9)^1 = (U^1 , U9), part (1) follows from 12.8.10.2. Then by (1)
and 12.8.10.2,
Zu n Z(I~) = Zu n Z[j n zf;. = Z(I2) n Z(I~):::; Cu(L) = 1,
since L = (L 2 , L&), and Cu(L) = 1 by 14.3.25. D
LEMMA 14.3.29. Assume there exists 1 -=/:-e E Z(J 2 ) n Z, and let Ve := (eL).
Then(1) Ve is of dimension 3, 4, 6, or 7, and Ve has an quotient L-module isomorphic
to the dual of V.
(2) J(T) :::1 LT.
PROOF. By 12.8.10.2, Z(J. 2 ) :::; Zu, so by choice of e and 14.3.25, [L, e] -=/:-1.
Thus I 2 T = CLr(e), so JeLTJ = 7. Thus (1) follows from H.5.3. As usual VVe E
R 2 (LT) by B.2.14, so as there is a quotient of Ve isomorphic to the dual of V as
an LT-module, (2) follows from Theorem B.5.6. DLEMMA 14.3.30. (1) IZ(I2)I:::; 2.
(2) If Z(I2) -=/:-1, then the image of Z(Ib) in iI is the subgroup of order 2generated by an involution of type a 2 in Sp(U) with [U, Z(Ib)] = V.
PROOF. We may assume Z(I 2 ) -=/:-1. By 14.3.28.1, Z(Ib) = Z[j n Zf;., so by
12.8.10.4,
[Z(Ib), W]:::; [Z[j, W]:::; ZuVi = ZuV[ and [Z(Jb), Un H^1 ]:::; ZuVf.
By 12.8.4.1 and G.2.5.1, [J = 02 (L 1 ), so U = Cu(V[)Cu(Vf) = W(U n H^1 ) from
the action of Lon V, and hence [U,Z(Ib)]:::; ZuV, with V ~ VZu/Zu of rank- Thus the image of Z(I~) in iI is either trivial, or is (a) of order 2, where a is the
element of Sp(U) of type a 2 with [U, a]= V, and in the latter case (2) holds. We will
show that Z(J~) is faithful on U. This will prove (1), and complete the proof of (2).
So let A := Cz(I~)(U); we must show A= 1. Applying 12.8.10.6 with Vi =
V1 Vj_9 and V1 Vf in the role of "Vi", A:::; V[ ZunV{ Zu = Zu, so A:::; ZunZ(Ib) = 1
by 14.3.28.2, completing the proof. DLEMMA 14.3.31. Z(h) = 1.
PROOF. Assume Z(I2) -=/:-1. Then by 14.3.30.1, Z(I 2 ) = (e) is of order 2, ande E Zu by 12.8.10.2. Further as T normalizes 12 , e E Z. Let a:= e^1. By 14.3.30.2,
a is the involution in Sp(U) of type a2 with [U, a] = v.
Let K := (aH). Then [a, Zu]:::; ZunZ(Ib) = 1by14.3.28.2. Thus a centralizes
Zu, so K does too. In particular (K,hT) :::; Ca(e) =:Ge. Also then CK(U) =
CK(U) = 02(K) using 12.8.4.4, so K/0 2 (K) ~ k ~ K*.
By 14.3.24, either U is the natural module for iI ~ G 2 (2), or d = 4 and one of
conclusions (i)-(iii) of 14.3.23.2 holds.Assume first that one of the cases other than case (i) of 14.3.23.2 holds. Then
F(H) is simple, so F(H) :::; k and K 1 := K^00 E C(H) with K 1 = F*(H). If