On common zeros of characters of finite groups

TL;DR

This paper investigates the structure of finite groups through the zeros of their irreducible characters, focusing on the minimal number of conjugacy classes and characters involved in vanishing phenomena.

Contribution

It introduces new bounds and characterizations for vanishing elements and conjugacy classes in finite groups, advancing understanding of their representation-theoretic properties.

Findings

Minimum conjugacy classes for all non-linear characters to vanish on one class

Minimum number of non-linear characters sharing a common zero

Structural implications for finite groups with specific vanishing properties

Abstract

Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $\chi\in\text{Irr}(G)$ such that $\chi(g)=0$ (i.e., $g$ is a zero of $\chi$) and, in this case, the conjugacy class $g^G$ of $g$ in $G$ is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group $G$ such that every non-linear $\chi\in\text{Irr}(G)$ vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.