Group Rings of Dihedral Groups

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


In this chapter we continue the study of stably free cancellation over the integral group rings Z[F n ×Φ] in the case where Φ is the dihedral group of order 2m defined by the presentation
$$D_{2m} = \langle x, y \vert x^m = y^2 = 1 , yx = x^{m-1}y \rangle.$$
Our main result, first proved in Johnson (Q. J. Math., 2011, doi: 10.1093/qmath/har006), is that Z[F n ×D 2p ] has SFC when p is an odd prime. This breaks down for p=2. Although Z[C ×D 4] still has SFC (the case n=1) when n≥2 a result of O’Shea shows that Z[F n ×D 4] has infinitely many isomorphically distinct stably free modules of rank 1.


Commutative Diagram Commutative Ring Group Ring Free Module Dihedral Group 
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Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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