GVZ-groups, Flat Groups, and CM-Groups

TL;DR

This paper characterizes GVZ-groups as flat groups, relates their nilpotence class to irreducible character degrees, and links certain CM-groups to GVZ-groups with character values in the prime field.

Contribution

It establishes the equivalence between GVZ-groups and flat groups and connects properties of CM-groups to GVZ-group characteristics.

Findings

GVZ-groups are exactly flat groups

Nilpotence class of GVZ-groups is bounded by irreducible character degrees

Certain CM-groups are GVZ-groups with character values in the prime field

Abstract

We show that a group is a GVZ-group if and only if it is a flat group. We show that the nilpotence class of a GVZ-group is bounded by the number of distinct degrees of irreducible characters. We also show that certain CM-groups can be characterized as GVZ-groups whose irreducible character values lie in the prime field.