Abelian Sylow Subgroups In A Finite Group, Ii

Let p be a prime. We prove that Sylow p-subgroups of a finite group G are abelian if and only if the class sizes of the p-elements of G are all coprime to p, and, if p is an element of{3, 5}, the degree of every irreducible character in the principal p-block of G is coprime to p. This gives a complete solution to a problem posed by R. Brauer in 1963. (C) 2014 Elsevier Inc. All rights reserved.