# Integral group rings of solvable groups with trivial central units

## TL;DR

This paper characterizes finite solvable groups with integral group rings having only trivial central units, showing their order's prime divisors are limited to 2, 3, 5, and 7, and classifies related Frobenius groups.

## Contribution

It establishes prime divisor restrictions for solvable groups with trivial central units and classifies Frobenius groups with this property.

## Key findings

- Prime divisors of such groups are only 2, 3, 5, and 7.
- Links between trivial central units and inverse semi-rational groups.
- Classification of Frobenius groups with trivial central units.

## Abstract

The integral group ring $\mathbb{Z} G$ of a group $G$ has only trivial central units, if the only central units of $\mathbb{Z} G$ are $\pm z$ for $z$ in the center of $G$. We show that the order of a finite solvable group $G$ with this property, can only have $2$, $3$, $5$ and $7$ as prime divisors, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.