An inequality for finite permutation groups

JOURNAL

OF COMBINATORIAL

Series A 27, 119 (1979)

THEORY,

Note An Inequality

for Finite

Permutation

Groups

MASAO KIYOTA Department of Mathematics, Faculty of Science, University of Tokyo, Hongo, Tokyo, 113, Japan Communicated by the Managing Editors

ReceivedOctober20, 1977

An inequalitywhichhasa combinatorialtheoreticalflavor is provedfor finite permutationgroups.

In a combinatorics and group theory seminar at Gakushuin University, E. Bannai and M. Deza posed us the following conjecture: let G be a permutation group of degree n and let { f( g) I g E G, g # l} = {& ,I, ,..., &.} where f( g) denotes the number of fixed points by g E G. Then 1G 1 < n:-, (n - &). Here, we prove this conjecture. Namely, we have THEOREM. Let G be a permutation group of degreen, and let Then 1G 1divides nisi (n - lJ.

(f ( g) 1g EG,

g f 11= K , 12,.">l,}.

Proof: Let 19be the permutation character of G, i.e., e(g) = let lc be the identity character of G. Then it is well known that

f(g), and

65:= fi (e - Ii. IG) i=l

is a generalized character of G, namely, 6 is a Z-linear combination of irreducible characters of G. By the definition of 0, we have that 8(g) = 0 for all g E G, g # 1. Hence, the multiplicity of lG in 6 is given by

Thus, we get the desired result.

119 0097-3165/79/040119-01$02.00/0 Copyright 0 1979 by AcademicPress, Inc. All rights of reproduction in any form reserved.