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Group Rings of Quaternion Groups

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

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.

Keywords

Generalized Quaternion Group Fibre Square Ring Involution Karoubi Principal Ideal Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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    O’Meara, O.T.: Introduction to Quadratic Forms. Springer, Berlin (1963) Google Scholar
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    Samuel, P.: Algebraic Theory of Numbers. Kershaw, London (1972) Google Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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