Coherent groups of units of integral group rings and direct products of free groups

Abstract

AbstractWe classify the finite groups G for which $\mathcal{U}({\mathbb Z} G)$, the group of units of the integral group ring of G, does not contain a direct product of two non-abelian free groups. This list of groups contains all the groups for which $\mathcal{U}({\mathbb Z} G)$ is coherent. This reduces the problem to classify the finite groups G for which $\mathcal{U}({\mathbb Z} G)$ is coherent to decide about the coherency of a finite list of groups of the form SLn(R), with R an order in a finite dimensional rational division algebra.