Recognising Abelian Sylow Subgroups in Finite Groups

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].