Group Rings of Quaternion Groups

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


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.


Generalized Quaternion Group Fibre Square Ring Involution Karoubi Principal Ideal Domain 
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  1. 45.
    Hurwitz, A.: Über die Zahlentheorie der Quaternionen. Collected Works (1896) Google Scholar
  2. 61.
    Kamali, P.: Stably free modules over infinite group algebras. Ph.D. Thesis, University College London (2010) Google Scholar
  3. 67.
    Lam, T.Y.: Serre’s Problem on Projective Modules. Springer, Berlin (2006) Google Scholar
  4. 77.
    O’Meara, O.T.: Introduction to Quadratic Forms. Springer, Berlin (1963) Google Scholar
  5. 83.
    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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