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.

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.