Zeros of primitive characters of finite groups

TL;DR

This paper classifies certain finite non-solvable groups based on the unique vanishing conjugacy class of a faithful primitive irreducible complex character, extending classical results on character zeros.

Contribution

It provides a complete classification of non-solvable groups with a faithful primitive irreducible character vanishing on exactly one conjugacy class, answering a question by Dixon and Rahnamai Barghi.

Findings

Classification of specific non-solvable groups based on character zeros

Extension of Burnside's theorem on zeros of characters

New insights into the structure of groups with unique character zero

Abstract

We classify finite non-solvable groups with a faithful primitive irreducible complex character that vanishes on a unique conjugacy class. Our results answer a question of Dixon and Rahnamai Barghi and suggest an extension of Burnside's classical theorem on zeros of characters.