On vanishing criteria that control finite group structure

TL;DR

This paper explores how arithmetic conditions on vanishing conjugacy classes, where characters evaluate to zero, influence the structure of finite groups, extending classical results to these special classes.

Contribution

It demonstrates that classical structural results for finite groups remain valid when considering only vanishing conjugacy classes.

Findings

Vanishing conjugacy classes can determine group structure.

Arithmetic conditions on these classes influence group properties.

Classical results extend to vanishing classes.

Abstract

Many results have been established that show how arithmetic conditions on conjugacy class sizes affect group structure. A conjugacy class in $G$ is called vanishing if there exists some irreducible character of $G$ which evaluates to zero on the conjugacy class. The aim of this paper is to show that for some classical results it is enough to consider the same arithmetic conditions on the vanishing conjugacy classes of the group.