# Unit groups of integral finite group rings with no noncyclic abelian finite subgroups

## TL;DR

This paper investigates the structure of units in integral group rings, establishing conditions under which certain noncyclic subgroups exist, linking properties of units to the original group structure.

## Contribution

It proves that noncyclic subgroups of order p^2 in units correspond exactly to those in the original group, extending known results for p=2.

## Key findings

- Noncyclic subgroups of order p^2 in units only exist if they exist in G for odd p.
- For p=2, the Brauer--Suzuki theorem applies, confirming the correspondence.
- The results connect the subgroup structure of units with that of the original finite group.

## Abstract

It is shown that in the units of augmentation one of an integral group ring $\mathbb{Z} G$ of a finite group $G$, a noncyclic subgroup of order $p^{2}$, for some odd prime $p$, exists only if such a subgroup exists in $G$. The corresponding statement for $p=2$ holds by the Brauer--Suzuki theorem, as recently observed by W. Kimmerle.