Rendiconti del Seminario Matematico della Università di Padova


Full-Text PDF (82 KB) | Metadata | Table of Contents | RSMUP summary
Volume 133, 2015, pp. 117–123
DOI: 10.4171/RSMUP/133-6

Groups having complete bipartite divisor graphs for their conjugacy class sizes

Roghayeh Hafezieh[1] and Pablo Spiga[2]

(1) Department of Mathematics, Gebze Institute of Technology, Gebze, Turkey
(2) Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via Cozzi, 55, 20125, Milano, Italy

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].

Keywords: Bipartite divisor graph, conjugacy class size, extra-special group

Hafezieh Roghayeh, Spiga Pablo: Groups having complete bipartite divisor graphs for their conjugacy class sizes. Rend. Sem. Mat. Univ. Padova 133 (2015), 117-123. doi: 10.4171/RSMUP/133-6