# Characterizations of nested GVZ-groups by central series

## TL;DR

This paper characterizes nested GVZ-groups, a special class of nilpotent groups, through specific ascending and descending central series, enhancing understanding of their structural properties.

## Contribution

It provides new characterizations of nested GVZ-groups using particular central series, linking character theory with group structure.

## Key findings

- Nested GVZ-groups characterized by specific central series
- Connection between character properties and group central series
- Enhanced understanding of nilpotent group structures

## Abstract

Many properties of groups can be defined by the existence of a particular normal series. The classic examples being solvability, supersolvability and nilpotence. Among the nilpotent groups are the so-called nested GVZ-groups --- groups where the centers of the irreducible characters form a chain, and where every irreducible character vanishes off of its center. In this paper, we show that nested GVZ-groups can be characterized by the existence of a certain ascending central series, or by the existence of a certain descending central series.