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∈{3,5}p∈{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.