Some results on GVZ-groups with two character degrees

Abstract

We investigate the finite groups $G$ for which $\ddot{I}‡(1{)}^2=|G:Z(\ddot{I}‡)|$ for all characters $\ddot{I}‡\in Irr(G)$ and $|cd(G)|=2$, where $cd(G)=\{\ddot{I}‡(1)|\ddot{I}‡\in Irr(G)\}$. 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 $\{1,p\}$.