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 $G\setminus \mathbf{Z}(G)$ (where $\mathbf{Z}(G)$ 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$(p,n)\ne 1$.

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].