Group Rings of Quaternion Groups

Abstract

In this chapter we extend the study of stably free cancellation for Z[F _n ×Φ] to the cases where Φ is the quaternion group Q(4m) of order 4m defined by the presentation

$$Q(4m) = \langle x , y \vert x^m = y^2 , xyx = y\rangle.$$

Here we find a marked contrast with the dihedral and cyclic cases. We first show by a delicate calculation that Z[C _∞×Q(8)] has infinitely many distinct stably free modules of rank 1. Whilst this result might seem unduly specific, it nevertheless implies a similar conclusion for Z[F _n ×Q(8m)] whenever m,n≥1. We conclude with a brief survey of what is known for the group rings Z[F _n ×Q(4m)] when m is odd.