Non-solvable groups and the two-prime hypothesis on conjugacy class sizes

Abstract

For a finite group G, $c{s}^{\ast }(G)$ denotes the set of non-trivial conjugacy class sizes of G. The group G is said to satisfy the two-prime hypothesis if, for every a, $b\in c{s}^{\ast }(G)$, the number of primes dividing the greatest common divisor of a and b is at most two. The aim of this article is to investigate the simple groups satisfying the two-prime hypothesis. Also, as a result, some non-solvable groups satisfying the two-prime hypothesis are studied.

Keywords:

• Conjugacy classes
• non-solvable groups
• two-prime hypothesis

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20E45
• Secondary: 20D05
• 05C25

Acknowledgments

The authors would like to express their gratitude to the referee for careful reading and several useful comments, which improved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.