# On zeros of irreducible characters lying in a normal subgroup

## TL;DR

This paper investigates the properties of elements in a normal subgroup of a finite group that do not vanish under any irreducible character, establishing conditions for the existence of normal Sylow p-subgroups and exploring related arithmetical properties.

## Contribution

It proves that if all p-elements of a normal subgroup are non-vanishing, then the subgroup has a normal Sylow p-subgroup, providing new insights into character zeros and conjugacy class sizes.

## Key findings

- Normal Sylow p-subgroup existence under non-vanishing p-elements
- Character zeros influence conjugacy class size properties
- New results for the case when the subgroup is the entire group

## Abstract

Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $\chi(g)\neq 0$ 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.