2.4. THE CASE WHERE r~ IS NONEMPTYLEMMA 2.4.20. Assume Z(L) =f 1. Then for x ET-S:
(1) B = D x DX is regular on 6. := Z(R) - (Z u zx).535(2) For u E 6., u is 2-central in Mand hence 2-central in G, Ca(u)::; M, and
uG nZ= 0.
(3) All involutions in Rare fused to u E 6. or z E z#.(4) R '.SJ M, so R '.SJ Co(u) foru E 6..
(5) For z E z#, Sylow 2-subgroups of Co(z) are in s^0.
(6) Ifu E 6. and X = (zo nX) is a 2-subgroup ofCo(u), then X _::; R.PROOF. As Z(L) =f 1, E2n ~ Z(L) = Z. By 2.4.11.3, Z is a TI-subgroup of G
with No(Z) = GQ, so for z E z#, Co(z) ::; GQ. Further SE Syl2(GQ) by 2.4.3,
so (5) holds and z is not 2-central in G.
By 2.4.8.6, B = D x Dx, while Z(R) = Z x zx by 2.4.16.2. By 2.4.8.4, Dx
is regular on z#, so as x interchanges D and Dx and Z and zx, D is regular on
(zx)#. Thus Z = Cz(R)(D), completing the proof of (1). Next Q and Qx are
the maximal elementary abelian subgroups of R by 2.4.8.7, while all elements of Q
are fused into Z(R) under L, so (3) holds. Then as z is not 2-central in G, but
Z x zx = Z(R) '.SJ T since T normalizes R by 2.4.8.1, u E Z(T) for some u E 6..
So as M = !M(T), Gu:= Co(u) ::; M, and then (2) follows from the transitivity
of Don 6. in (1).
Next we prove (4). Set P := 02 (M). As R = J(T) by 2.4.8.3, it suffices to show
that R::; P, since then R = J(P) by B.2.3.3. As F*(M) = P, u E CM(P) = Z(P),
so by (1), Z(R) = (uBT) ::; Z(P). Let W := (zo n P). By 2.4.19, W ::; R, so
as Bis irreducible on Q/Z(R), either W = Z(R) or W = R. Since W '.SJ M, (4)
holds if W = R. If W =f R then Z(R) = W '.SJ M so that M = No(Z(R)) since
M E M. But then as Z is a TI-subgroup of G, it follows from (1) and (2) that
M = NM(Z)(x). Now No(Z) = GQ = No(L) by 2.4.11, so NM(Z) normalizes
02 (NMnL(Z)) = R. As x also normalizes R, we conclude (4) holds in this case
also.
Finally assume the hypotheses of (6). Then X::; Co(u)::; M by (2), and as X
is a 2-group, X ::; Tm for some m EM. Then X ::; (z^0 n Tm) =Rm by 2.4.19, so
that X::; R by (4). D
In the r~mainder of the treatment of the cases = 1, we let z denote an involution
of z#. If Z(L) =f 1, let u denote an element of the set 6. defined in 2.4.20.1.
LEMMA 2.4.21. (1) R is the strong closure of Q in T.
(2) i^0 n T ~ R for each involution i in R.
PROOF. By parts (2) and (7) of 2.4.8, all involutions in Rare fused into Q, so
(1) implies (2).By 2.4.19, R is contained in the strong closure of Q in T. Hence we may
assume that a is an involution in T - R fused into Q, and it remains to derive a
contradiction. If Z(L) = 1 then Lis transitive on Q#, so a= z^9 for some g E G.
If Z(L) =f 1 then by 2.4.20.3, either a= z9, or a= u9 for u E 6.. Set I:= CR(a)
and let I::; T E Syl 2 (Co(a)) and set R := J(T*).
We claim that if Z(L) =f 1 then a E S. Thus we assume Z(L) =f 1 and
a ET - S, and it remains to derive a contradiction. By 2.4.14, I is of type Sz(q),
so the involutions of I lie in 6. rather than in Z or zx = za, since a E T - S.
Assume first that a= z9. By 2.4.20.5, T E S^0 , and by 2.4.12.1, T / R* is cyclic,