Average number of zeros of characters of finite groups

TL;DR

This paper investigates the average number of zeros in the character table of finite groups, establishing thresholds that characterize group properties such as solvability, supersolvability, and abelianness.

Contribution

It introduces the invariant anz(G) and provides thresholds for solvability, supersolvability, and abelianness based on its value, offering new characterizations of finite groups.

Findings

anz(G) < 1 implies G is solvable

anz(G) < 1/2 implies G is supersolvable

anz(G) < 1/3 characterizes abelian 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.