Global and local properties of finite groups with only finitely many central units in their integral group ring

TL;DR

This paper investigates finite groups with only trivial central units in their integral group rings, analyzing their properties, character tables, and classifying simple cut groups, revealing that such groups are more common than previously thought.

Contribution

It provides new criteria for nilpotent groups of class 2 to be cut and offers a complete classification of simple cut groups, expanding understanding of their structure.

Findings

Number of rational-valued irreducible characters equals the number of rational-valued conjugacy classes.

Criteria established for nilpotent groups of class 2 to be cut.

Complete list of simple cut groups provided.

Abstract

The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same, in particular the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.