Partial GVZ-groups

TL;DR

This paper investigates a new class of groups with a normal subgroup where all irreducible characters either contain this subgroup in their kernel or have a specific vanishing property, revealing limited and structured forms.

Contribution

It introduces and studies groups with a normal subgroup where characters are either kernel-contained or of central type, expanding understanding of GVZ-like groups.

Findings

The structure of these groups is surprisingly limited.

They share properties with both central type and GVZ-groups.

The class of such groups exhibits constrained and well-defined characteristics.

Abstract

Following the literature, a group $G$ is called a group of central type if $G$ has an irreducible character that vanishes on $G\setminus Z(G)$. Motivated by this definition, we say that a character $\chi\in {\rm Irr}(G)$ has central type if $\chi$ vanishes on $G\setminus Z(\chi)$, where $Z(\chi)$ is the center of $\chi$. Groups where every irreducible character has central type have been studied previously under the name GVZ-groups (and several other names) in the literature. In this paper, we study the groups $G$ that possess a nontrivial, normal subgroup $N$ such that every character of $G$ either contains $N$ in its kernel or has central type. The structure of these groups is surprisingly limited and has many aspects in common with both central type groups and GVZ-groups.