The cancellation property for projective modules over integral group rings

TL;DR

This paper investigates when projective modules over integral group rings exhibit cancellation properties, providing a classification of finite groups based on this criterion and extending understanding of module theory over such rings.

Contribution

It offers a partial classification of finite groups for which the integral group ring has projective cancellation, introducing a new cancellation theorem and applying existing results for a comprehensive analysis.

Findings

Identifies groups with and without projective cancellation property.

Proves a new cancellation theorem based on a relative Eichler condition.

Classifies finite groups where projective cancellation fails or holds.

Abstract

We obtain a partial classification of the finite groups $G$ for which the integral group ring $\mathbb{Z} G$ has projective cancellation, i.e. for which $P \oplus \mathbb{Z} G \cong Q \oplus \mathbb{Z} G$ implies $P \cong Q$ for projective $\mathbb{Z} G$-modules $P$ and $Q$. In particular, we determine when projective cancellation holds for a finite group with no exceptional binary polyhedral quotients. To do this, we prove a cancellation theorem based on a relative version of the Eichler condition. We then use a group theoretic argument to precisely determine the class of groups not covered by this result. The final classification is then obtained by applying results of Swan, Chen and Bley-Hofmann-Johnston which show failure of projective cancellation for certain groups.