On the Congruence Subgroup Problem for integral group rings

TL;DR

This paper investigates the Congruence Subgroup Problem for units in integral group rings, providing partial classifications and conditions under which the associated congruence kernel is finite or infinite.

Contribution

It offers an approximation to classifying finite groups based on the finiteness of the congruence kernel of their unit groups, identifying key group families and open problems.

Findings

Identifies three families of finite groups related to the congruence kernel finiteness.

Provides a list of 19 specific groups linked to the kernel's infiniteness.

Highlights the difficulty of determining the kernel's finiteness for certain algebraic structures.

Abstract

Let $G$ be a finite group, $\Z G$ the integral group ring of $G$ and $\U(\Z G)$ the group of units of $\Z G$. The Congruence Subgroup Problem for $\U(\Z G)$ is the problem of deciding if every subgroup of finite index of $\U(\Z G)$ contains a congruence subgroup, i.e. the kernel of the natural homomorphism $\U(\Z G) \rightarrow \U(\Z G/m\Z G)$ for some positive integer $m$. The congruence kernel of $\U(\Z G)$ is the kernel of the natural map from the completion of $\U(\Z G)$ with respect to the profinite topology to the completion with respect to the topology defined by the congruence subgroups. The Congruence Subgroup Problem has a positive solution if and only if the congruence kernel is trivial. We obtain an approximation to the problem of classifying the finite groups for which the congruence kernel of $\U(\Z G)$ is finite. More precisely, we obtain a list $L$ formed by three families of finite groups and 19 additional groups such that if the congruence kernel of $\U(\Z G)$ is infinite then $G$ has an epimorphic image isomorphic to one of the groups of $L$. About the converse of this statement we at least know that if one of the 19 additional groups in $L$ is isomorphic to an epimorphic image of $G$ then the congruence kernel of $\U(\Z G)$ is infinite. However, to decide for the finiteness of the congruence kernel in case $G$ has an epimorphic image isomorphic to one of the groups in the three families of $L$ one needs to know if the congruence kernel of the group of units of an order in some specific division algebras is finite and this seems a difficult problem.