. 2.5. ELIMINATING THE SHADOWS WITH r~ EMPTY 559
(ii) FI~ Aut(L3(3)) and Cs(Kz) = Sc(y), where y induces an outer automor-
phism on K with fJ2 = z.Recall from Notation 2.5.9 that we may choose (U, Hu) E U*(H) with U:::; S.
By 2.5.12.3, Hu = NH(E) for some 4-subgroup E of SK and U = Ns(E) E
Syl2(Hu ). Then as 02 (Hu) ~ A4 by 2.5.12.2, QE := 02 (Hu) = Cs(E). In case (i)
S induces inner automorphisms on K, so S =Sc x SK, and hence as E = CK(E) by
2.5.12.2, QE =Sc x E. On the other hand in case (ii), we compute that e EE-(z)
inverts y, so QE =(Sc x E)(f), where f = yk and k is one of the two elements of
02(Kz) of order 4 inverted by e.
Recall x E Nr(S) n Cr(Kz), so x normalizes Cs(Kz), and hence[e, x] E Sn Cr(Kz) = Cs(Kz).
But if case (i) holds then Cs(Kz) = Sc(z) :::; QE, and by the previous para-
graph ScE = QE, so x E Na(QE)· Then U < Ns(QE)(x):::; Na(QE), contradict-
ing 2.5.12.4.
Therefore case (ii) holds. Here x normalizes Cs(Kz) = Sc(Y), while ScnS 0 =
1 by 2.5.5.1, so as t E Sc, Sc is cyclic of order 2 or 4.
Assume Sc ~ Z4. Then by 2.5.5.2, ScS 0 = Sc x S 0 , so as 'fl and Sc are
of order 4, Cs(Kz) = Sc x S 0 is abelian. In particular y centralizes Sc, so sinceS = ScSK(y), Z(S) contains Sc ~ Z4, contrary to 2.5.11.1..
Therefore Sc = (t), so Cs(Kz) = (t, y), and as y^2 = z, y^2 = z or tz. Hence
as we saw tx = tz, while x normalizes if!(Sc(y)) = (y^2 ), y^2 = z. Therefore as
H=KS,
H = (t) x A,where A:= K(y) ~ Aut(L 3 (3)). Observe that Sc(z) = (t, z) = Z(S) using 2.5.11.1.
Assume that [e, x] E (t, z). Then as x acts on Z(S) = (z), x acts on ScE ::::l Hu,
so that Ns(E) < Ns(E)(x) :::; Na(ScE), again contrary to 2.5.12.4. Therefore
[e,x] (j_ (t,z).
Next A is transitive on involutions in A - K, and on Es-subgroups of A, with
representatives f and F := (f, E), respectively. Further we may choose notation so
that CA(!)= (f) x CK(!) with CK(i) = NK(E) ~ S4. Now x acts on Cs(Kz) =
(t, y), and we've seen that [e, x] E Cs(Kz) - (t, z), so replacing y by a suitable
element of y(t, z), we may take ex = ey. Thus ey E A - K is an involution in
ea = za, so all involutions in p# are in za. On the other hand, we saw that
tz = tx E ta, so all involutions in tK are in ta, and in particular te E t^0. Further
(te)x = txex = tzey = tey-^1 ,
with ey-^1 an involution in A - K; so all involutions in H - A are in t^0.
As pA is the set of Es-subgroups of A, and QE = 02(NH(E)) = (t) x F,
Q}i is the set of E 16 -subgroups of H. By 2.5.11.2, Gt E I'* and S E Syl 2 (Gt)·
So (t) is Sylow in Ca. (K), and hence using Cyclic Sylow-2 Subgroups A.1.38 we
conclude that Ca.(K) = O(Gt)(t). We saw that K ::::l Gt so z EK:::; C(O(Gt)).
Thus O(Gt) = 1 since z inverts O(Gt) by 2.3.9.5. Hence Gt= KS= H. Therefore
Ca(t) =His transitive on its E 16 -subgroups with representative QE, so by A.1.7.1,
Na(QE) is transitive on t^0 nQE = QE-F of order 8. Then INa(QE): Na.(QE)I =
8, whereas Ns(E) E Syb(Na(QE)) by 2.5.12.4, and Ns(E):::; Gt. Hence the proof
of 2.5.15 is at last complete. D