Determination of the stably free cancellation property for orders

TL;DR

This paper develops practical algorithms to determine whether orders in semisimple algebras have stably free cancellation, and applies these to classify certain finite groups with this property.

Contribution

It introduces algorithms for checking stably free cancellation in orders and classifies all groups of order up to 383 with this property.

Findings

Identified all finite groups of order ≤ 383 with SFC in their integral group rings.

Provided algorithms for practical determination of SFC in orders.

Enhanced understanding of the structure of orders with SFC.

Abstract

Let $K$ be a number field, let $A$ be a finite-dimensional semisimple $K$-algebra, and let $\Lambda$ be an $\mathcal{O}_{K}$-order in $A$. We give practical algorithms that determine whether $\Lambda$ has stably free cancellation (SFC). As an application, we determine all finite groups $G$ of order at most $383$ such that the integral group ring $\mathbb{Z}[G]$ has SFC.