Proportions of vanishing elements in finite groups

TL;DR

This paper investigates the proportion of vanishing elements in finite groups, establishing bounds for non-abelian groups and symmetric groups, and classifying groups that attain these bounds.

Contribution

It provides new bounds on the proportion of vanishing elements in finite groups and classifies groups that reach these bounds.

Findings

Proportion of vanishing elements in non-abelian groups is at least 1/2.

Symmetric groups of degree ≥ 5 have a vanishing element proportion of at least 2327/2520.

The bounds are proven to be optimal.

Abstract

In this paper, we study the proportion of vanishing elements of finite groups. We show that the proportion of vanishing elements of every finite non-abelian group is bounded below by $1/2$ and classify all finite groups whose proportions of vanishing elements attain this bound. For symmetric groups of degree at least $5$, we show that this bound is at least $2327/2520$ which is best possible.