Finite Groups with Four Conjugacy Class Sizes

Abstract

We determine the structure of all finite groups with four class sizes when two of them are coprime numbers larger than 1. We prove that such groups are solvable and that the set of class sizes is exactly {1, m, n, mk}, where m, n > 1 are coprime numbers and k > 1 is a divisor of n.

Key Words:

• Class sizes
• Conjugacy
• Finite groups

2000 Mathematics Subject Classification:

• 20D20
• 20E45

ACKNOWLEDGMENTS

We would like to thank the referee for all his comments and suggestions, and we are also truly grateful to Professor Kazarin for the detailed information he provided us on his article [19]. This work was partially supported by Proyecto MTM2007-68010-C03-03, by Proyecto MTM2010-19938-C03-02 and by Proyecto GV-2009-021 MTM2007-68010-C03-03 and the first author is also supported by grant Fundació Caixa-Castelló P11B2008-09.

Dedicated to Professor M. J. Iranzo on the occasion of her retirement.

Notes

Communicated by A. Turull.