Determination of the stably free cancellation property for orders

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.

To appear in Transactions of the American Mathematical Society; 44 pages; v3 includes many revisions following referee report; v2 includes many revisions, including corrections to results on which integral group rings Z[G] have SFC or not