Some results on GVZ-groups with two character degrees

TL;DR

This paper studies a special class of finite groups called GVZ-groups with exactly two character degrees, providing new characterizations, bijections between character sets, and explicit constructions of such groups.

Contribution

It introduces new characterizations and constructions of GVZ-groups with two character degrees, expanding understanding of their structure and properties.

Findings

Established bijections between character sets of related groups

Provided alternative characterizations of GVZ-groups with two degrees

Constructed an example of a GVZ-group with character degrees {1, p}

Abstract

We investigate the finite groups $G$ for which $\chi(1)^{2}=|G:Z(\chi)|$ for all characters $\chi \in Irr(G)$ and $|cd(G)|=2$, where $cd(G)=\{\chi(1)| \chi \in Irr(G)\}$. We call such a group a GVZ-group with two character degrees. We establish bijections between the sets of characters of some groups obtained from a GVZ-group with two character degrees. Additionally we obtain some alternate characterizations of a GVZ-group with two character degrees and we construct a GVZ-group having the character degree set $\{1,p\}$.