2. PERMUTATION GROUPS 5
By induction Nxa (S) is (k l)-transitive on fts {a}. The same argument applied
to a different element / 3 G A shows that Nx0(S) is (k l)-transitive on fts {ft}.
Since \fls\ 2, this implies the desired /c-transitivity of Nx(S) on fts-
The Bender-Suzuki Theorem SE relies for the final identification of the simple
groups on the following Background Result due to Suzuki [Su4].
THEOREM
2.5
(SUZUKI).
Let X be a 2-transitive permutation group on a set
ft of odd cardinality. Let a G ft and suppose that Xa has a nilpotent regular normal
subgroup of odd index. Then
O2'(X)
^
L2(2n), 2
£
2
(2t) or
U3(2n)
for some n2.
PROOF.
This is proved in [PI, Part II].
In the proof of Theorem SE, we shall also need at one point the following slight
generalization of Burnside's theorem on permutation groups of prime degree (cf.
[PI], [Hul; V.21.3], and Corollary 2.7 below).
THEOREM
2.6. Let X act on ft and suppose that K is a subgroup of X act-
ing regularly on ft {i.e., K is transitive on ft and \K\ = \ft\). Suppose further
that Cx(%) = K for every non-identity element x of K. Then either X acts 2-
transitively on ft or X = WNx(K) where W is the kernel of the X-action on
ft.
PROOF.
Set X = X/W. Then ~K ^ K and ~K C^(x) for all x G K. By [IG;
9.16], |C;x(x)| |Cx(#)| 1^1
f°r e v e r
y non-identity element x of K and so X
satisfies the hypotheses of the theorem.
If W zfc 1, we conclude by induction that either X acts 2-transitively or K X,
whence WK X. In the latter case since Cx(x) = K for all 1 7^ x G K, the group
WK is a Frobenius group with Probenius complement K. By the Schur-Zassenhaus
Theorem all complements to W in WK are VF-conjugate, so a Prattini argument
yields X = WKNX(K) = WNX(K) as desired.
Therefore we may assume that W 1. We set N NX(K), assume that
N X, and prove double transitivity.
Notice that by hypothesis, K is an abelian group. If g G X and K D
K9
^ 1,
then K = Cx(Kn
K9)
=
K9
and so g G N. In particular if K = N, then X is a
Probenius group with complement K. In that case, let Y be its Probenius kernel.
Then |y| = \X : K\ = \XQ\ for any a G ft. But then Y = Xa for all a G ft and so
Y = W 1 and X = K, contrary to assumption.
Thus we may assume that K N, whence iV is a Frobenius group with Probe-
nius kernel K. We have seen that K is a T.I. set in X, and so [Is; 7.18-7.20] applies
to yield the following results about the irreducible characters of X:
Either N is transitive on K# or there is a bijection \ ~* X* between the set of
irreducible characters \ of N such that K ^ ker x and the set £ of (exceptional)
irreducible characters x* of X which are not constant on K#, and this bijection
satisfies:
, , For any g E X not conjugate to any non-identity element of K, the
^ ^ value x*(#) is independent of the choice of x* £.
Now let 7r be the character afforded by the permutation representation of X on
ft. By [Is; 5.15], (71", lx) 1- Write n = lx + Si= i &
w
^
n e a c n
0i
a
nonprincipal
irreducible character of X. If £ = 1, then by [Is; 5.17], X is 2-transitive on ft and
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