A cancellation theorem for modules over integral group rings

TL;DR

This paper investigates when the integral group ring of a finite group has stably free cancellation, extending classical results and providing new conditions related to the group's algebraic structure.

Contribution

It extends Swan's results by establishing a new criterion for stable free cancellation in integral group rings, generalizing the Eichler condition.

Findings

SFC holds if at most one quaternion algebra appears in the Wedderburn decomposition.

Provides a new criterion for SFC based on the group's real representation structure.

Generalizes the Eichler condition for integral group rings.

Abstract

A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which the integral group ring $\mathbb{Z}G$ has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that $\mathbb{Z}G$ has SFC provided at most one copy of the quaternions $\mathbb{H}$ occurs in the Wedderburn decomposition of the real group ring $\mathbb{R}G$. This generalises the Eichler condition in the case of integral group rings.