Recognising Abelian Sylow Subgroups in Finite Groups

TL;DR

This paper characterizes finite groups with non-abelian Sylow p-subgroups where all p-elements have class size coprime to p, showing such groups contain specific simple subquotients and listing all such simple groups and primes.

Contribution

It proves the existence of simple subquotients with the same property in groups with non-abelian Sylow p-subgroups and class size conditions, completing a classification for all primes.

Findings

Identifies conditions under which groups contain simple subquotients with non-abelian Sylow p-subgroups.

Provides a complete list of simple groups and primes with the specified properties.

Solves a previously open problem for all primes, including p=3 and p=5.

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. This provides a solution to the problem which was left remaining after Tiep and Navarro established that is p is not equal to 3 or 5, then this can never happen [NT14].