Subgroup separability in integral group rings

TL;DR

This paper classifies finite groups based on whether their integral group rings have subgroup separability in the unit group, identifying all such groups except for two specific unresolved cases.

Contribution

It provides a comprehensive list of finite groups with subgroup separable unit groups in their integral group rings, highlighting the unresolved cases.

Findings

Identifies all finite groups with subgroup separable units in their integral group rings.

Pinpoints two specific group types where subgroup separability remains undecidable.

Advances understanding of the structure of units in integral group rings.

Abstract

We give a list of finite groups containing all finite groups $G$ such that the group of units $\Z G^*$ of the integral group ring $\Z G$ is subgroup separable. There are only two types of these groups $G$ for which we cannot decide wether $ZG^*$ is subgroup separable, namely the central product $Q_8 Y D_8$ and $Q_8\times C_p{with} p \text{prime and} p\equiv -1 \mod (8)$.