# Groups having complete bipartite divisor graphs for their conjugacy class sizes

### Roghayeh Hafezieh

Gebze Institute of Technology, Turkey### Pablo Spiga

Università degli Studi di Milano-Bicocca, Italy

## 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$(p,n)\neq 1$.

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].

## Cite this article

Roghayeh Hafezieh, Pablo Spiga, Groups having complete bipartite divisor graphs for their conjugacy class sizes. Rend. Sem. Mat. Univ. Padova 133 (2015), pp. 117–123

DOI 10.4171/RSMUP/133-6