Groups whose vanishing class sizes are not divisible by a given prime

Let G be a finite group. An element g is an element of G is a vanishing element of G if there exists an irreducible complex character chi of G such that chi(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.