Finite groups with non-trivial intersections of kernels of all but one irreducible characters

TL;DR

This paper characterizes finite groups where all but one irreducible character share a non-trivial kernel intersection, revealing a special class of groups with unique character table properties and extending previous classifications.

Contribution

It provides a complete characterization of finite groups with a specific kernel intersection property, including a subclass containing groups with non-linear characters of distinct degrees.

Findings

Groups with two character table columns differing by one entry exist.

Such groups have an irreducible character vanishing on all but two conjugacy classes.

The subclass includes all groups with non-linear characters of distinct degrees.

Abstract

In this paper we consider finite groups G satisfying the following condition: G has two columns in its character table which differ by exactly one entry. It turns out that such groups exist and they are exactly the finite groups with a non-trivial intersection of the kernels of all but one irreducible characters or, equivalently, finite groups with an irreducible character vanishing on all but two conjugacy classes. We investigate such groups and in particular we characterize their subclass, which properly contains all finite groups with non-linear characters of distinct degrees, which were characterized by Berkovich, Chillag and Herzog in 1992.