Group elements whose character values are roots of unity

TL;DR

This paper classifies all finite groups that have an element where every irreducible character evaluates to a root of unity, revealing a specific structural property related to character theory.

Contribution

It provides a complete classification of finite groups with an element whose irreducible characters are all roots of unity, a new characterization in group and character theory.

Findings

Identifies the structure of groups with such elements

Characterizes elements with root of unity values across all irreducible characters

Provides a classification theorem for these groups

Abstract

We classify all finite groups $G$ which possesses an element $x\in G$ such that every irreducible character of $G$ takes a root of unity value at $x$.