# Groups where the centers of the irreducible characters form a chain

## TL;DR

This paper studies groups with a chain-structured set of centers of irreducible characters, providing characterizations, structural insights, and classifications, including connections to nested GVZ groups and $p$-groups with specific character degrees.

## Contribution

It offers new characterizations of such groups, classifies groups with chain-structured kernels, and provides alternative proofs for existing theorems.

## Key findings

- Characterizations of groups with chain centers of irreducible characters
- Classification of groups with chain-structured kernels
- Alternative proof of Isaacs' theorem on $p$-groups with given character degrees

## Abstract

We consider groups where the centers of the irreducible characters form a chain. We obtain two alternate characterizations of these groups, and we obtain some information regarding the structure of these groups. Using our results, we are able to classify those groups where the kernels of the irreducible characters form a chain. We show that a result of Nenciu regarding nested GVZ groups is really a result about nested groups. We obtain an alternate proof of a theorem of Isaacs regarding the existence of $p$-groups with a given set of irreducible character degrees.