Group Rings of Cyclic Groups

  • F. E. A. JohnsonEmail author
Part of the Algebra and Applications book series (AA, volume 17)


In this chapter we begin the detailed study of the SFC property for group rings of the form Z[F n ×Φ] where F n is the free group of rank n≥1 and Φ is finite. In the first instance we consider the rings Z[F n ×C m ] where C m is the cyclic group of order m.

As Z[Φ] is a retract of Z[F n ×Φ] it follows that a prior condition for Z[F n ×Φ] to have stably free cancellation is that Z[Φ] should also have this property. The question of stably free cancellation for Z[Φ] and related rings has been studied extensively by Swan (Ann. Math. 76:55–61, 1962; K-Theory of finite groups and order (notes by E.G. Evans), Lecture notes in mathematics, vol. 149, Springer, Berlin, 1970; J. Reine Angew. Math. 342:66–172, 1983) and Jacobinski (Acta Math. 121:1–29, 1968), building upon earlier work of Eichler (Math. Z. 43:102–109, 1938).


Group Ring Jacobinski Related Rings Eichler Condition Quaternion Factor 
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© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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