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