Recognizing Abelian Sylow Subgroups in Finite Groups

Abstract

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p, then G contains a simple group as a subquotient which exhibits the same property. In addition, we provide a list of all the simple groups and primes such that the Sylow p-subgroups are non-abelian and all p-elements have class size coprime to p.

Key Words:

• Abelian Sylow subgroups

2010 Mathematics Subject Classification:

• Primary: 20D06
• Secondary: 20D20

ACKNOWLEDGMENTS

We would like to thank Ron Solomon for useful discussions we had at the final stages of writing this paper. In addition, we would also like to thank the referee for providing thorough comments and clearer ways of presenting the results given in this article.

Notes

Communicated by P. Tiep.