ARITHMETICAL CONDITIONS ON THE LENGTH OF THE CONJUGACY CLASSES OF A FINITE-GROUP

This paper is a contribution to the study of the influence on the structure of a finite group G of some arithmetical conditions imposed an the length of the conjugacy classes of G. Our aim is to point out conjugacy-class analogues of some of the many results that in the last twenty years have been discovered on the relations between the structure of a group G and the arithmetical structure of the set of degrees of the irreducible complex characters of G. See [10] for a thorough survey on these topics. In most cases a surprising parallelism has been observed, whose grounds are not yet well understood, of results in the conjugacy-class and in the character-degree context. We refer also to the recent works [2-8]. Our interest in this direction was stimulated when, while visiting the University of Mainz in winter 1989, we were able to read a preliminary version of paper [6] by D. Chillag and M. Herzog. We consider first ''coprimity''-conditions on the lengths of the conjugacy classes of a finite group G (Theorem 4, Corollaries 9 and 12). The introduction of a graph associated with the conjugacy classes allows in this situation a vivid visual realization. Theorem 13, which is an improvement in the soluble case of a previous result by N. Ito, Corollary 14, and the ''Three-vertex'' Theorem 16 point out the ''richness,'' in the sense of abundance of edges, of the above-mentioned conjugacy-class graph. Further, in the soluble case, Theorem 17 gives a best-possible bound for the diameter of the graph.