Groups having complete bipartite divisor graphs for their conjugacy class sizes

Abstract

Given a finite group G, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G\Z(G) (where Z(G) denotes the centre of G) and the set of prime numbers that divide these conjugacy class sizes, and with {p, n} being an edge if gcd(p, n) not equal 1. In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K-2,(5), giving a solution to a question of Taeri [15].