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A comparison of group algebras of dihedral and quaternion groups

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Abstract

The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly, the semisimple case (where the characteristic of the field is greater than 2) is considered and the minimum noncommutative counterexample to the Isomorphism Problem is identified.

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Acknowledgements

The authors wish to acknowledge useful conversations with Fergal Gallagher and Yuval Ginosar.

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Authors and Affiliations

  1. Institute of Technology Sligo, Sligo, Ireland

    Leo Creedon & Kieran Hughes

  2. Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY, 40475, USA

    Steve Szabo

Authors
  1. Leo Creedon
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  2. Kieran Hughes
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  3. Steve Szabo
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Correspondence to Steve Szabo.

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Creedon, L., Hughes, K. & Szabo, S. A comparison of group algebras of dihedral and quaternion groups. AAECC 32, 245–264 (2021). https://doi.org/10.1007/s00200-020-00485-1

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  • DOI: https://doi.org/10.1007/s00200-020-00485-1

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