Conjugacy class lengths of finite groups with disconnected prime graph

Abstract

For a nite group G, we write c (G) to denote the prime divisor set of the various conjugacy class lengths of G and c (G) the maximum number of distinct prime divisors of a single conjugacy class length ofG. It is conjectured that the inequalityjc (G)j 3c (G) is true for every nite group. In this note, we prove that the inequality jc (G)j 2c (G) is true when the ( element order ) prime graph ( G) is disconnected by using Gruenberg-Kegel classication theorem and Casolo’s result. Mathematics Subject Classication: 20Dxx (Primary), 20E15 (Secondary)