Integral group rings with all central units trivial

TL;DR

This paper characterizes finite groups whose integral group rings have only trivial central units, focusing on prime-power and metacyclic groups, and classifies metacyclic groups based on the central height of their unit groups.

Contribution

It provides a complete classification of prime-power and metacyclic groups with trivial central units in their integral group rings and analyzes the central height in metacyclic groups.

Findings

Prime-power groups with trivial central units are characterized.

Metacyclic groups with this property are classified.

The central height of unit groups in metacyclic groups is determined.

Abstract

The object of study in this paper is the finite groups whose integral group rings have only trivial central units. Prime-power groups and metacyclic groups with this property are characterized. Metacyclic groups are classified according to the central height of the unit groups of their integral group rings.