Average number of zeros of characters of finite groups

Abstract

There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by $\mathrm{anz}(G)$, the average number of zeros of characters of a finite group $G$ as the number of zeros in the character table of $G$ divided by the number of irreducible characters of $G$. We show that if $\mathrm{anz}(G)<1$, then the group $G$ is solvable and also that if $\mathrm{anz}(G)<\frac{1}{2}$, then $G$ is supersolvable. We characterise abelian groups by showing that $\mathrm{anz}(G)<\frac{1}{3}$ if and only if $G$ is abelian.

10 pages