# On vanishing class sizes in finite groups

## TL;DR

This paper investigates the arithmetic properties of vanishing conjugacy class sizes in finite groups, focusing on elements with zero character values and their implications for group structure.

## Contribution

It introduces new arithmetical insights into the sizes of vanishing conjugacy classes and their role in understanding finite group properties.

## Key findings

- Characterizes conditions for vanishing elements in finite groups
- Establishes relationships between vanishing class sizes and group structure
- Provides criteria for the existence of vanishing conjugacy classes

## Abstract

Let $G$ be a finite group. An element $g$ of $G$ is called a vanishing element if there exists an irreducible character $\chi$ of $G$ such that $\chi(g) = 0$; in this case, we say that the conjugacy class of $g$ is a vanishing conjugacy class. In this paper, we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group.