Finite groups whose prime graph on class sizes is a block square

Abstract

Let G be a finite group, and let $\Delta (G)$ be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are the prime numbers dividing some conjugacy class size of G, and two distinct vertices p, q are adjacent if and only if pq divides some class size of G. In this paper, we characterize the structure of those groups G whose prime graph $\Delta (G)$ is a block square.

Keywords:

• Finite groups
• conjugacy classes
• prime graph

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20E45

Acknowledgements

This research has been carried out during a stay of the author at the Dipartimento di Matematica e Informatica “Ulisse Dini” (DIMAI) of Università degli Studi di Firenze. He wishes to thank the members of the DIMAI for their hospitality, and Centro Universitario EDEM for its support.

Additional information

Funding

This research is supported by Proyecto PGC2018-096872-B-I00 from the Ministerio de Ciencia, Innovación y Universidades (Spain), and Proyecto AICO/2020/298 from the Generalitat Valenciana (Spain).