Proportions of vanishing elements in finite groups

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.