Finite Groups with few character zeros

Abstract

Let G be a finite group, and let anz(G) be the average number of zeros of characters of G, that is the number of zeros in the character table of G divided by the number of irreducible characters of G. In this paper, we classify the finite groups G with anz(G)≤1. In particular, we show that for a rational number a≤1, there exists a finite group G with anz(G)=a if and only if a∈{0,23,45,1}, or a=kk+2 for some positive integer k.