On zeros of irreducible characters lying in a normal subgroup

Abstract

Let N be a normal subgroup of a finite group G. In this paper, we consider the elements g of N such that $$\chi (g)\ne 0$$ for all irreducible characters $$\chi$$ of G. Such an element is said to be non-vanishing in G. Let p be a prime. If all p-elements of N satisfy the previous property, then we prove that N has a normal Sylow p-subgroup. As a consequence, we also study certain arithmetical properties of the G-conjugacy class sizes of the elements of N which are zeros of some irreducible character of G. In particular, if $$N=G$$ , then new contributions are obtained.