Two dual questions on zeros of characters of finite groups

Abstract

For a finite group G and a non-linear irreducible complex character χ of G write υ(χ) = {g ∈ G|χ(g) = 0}. There are two dual questions as follows: what are the finite groups G such that υ(χ) is a conjugacy class for all but one of the non-linear irreducible characters χ of G, and what are the finite groups G whose character table has the property that all but one of the columns have at most one zero entry. We completely answer the two questions using the classification of finite simple groups.