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Finite groups with an automorphism that is a complete mapping

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Abstract

We show that a finite group G admitting an automorphism \(\alpha \) such that the function \(G\rightarrow G\), \(g\mapsto g\alpha (g)\), is bijective is necessarily solvable.

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Acknowledgements

The author would like to thank Arne Winterhof for suggesting to work on complete mappings on groups. Moreover, he would like to express his gratitude towards Peter Cameron, Michael Giudici, Laszlo Merai, Cheryl Praeger, Csaba Schneider and Arne Winterhof for some helpful comments during the work on this paper.

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  1. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Altenberger Straße 69, 4040, Linz, Austria

    Alexander Bors

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  1. Alexander Bors
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Correspondence to Alexander Bors.

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Communicated by John S. Wilson.

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The author is supported by the Austrian Science Fund (FWF), project J4072-N32 “Affine maps on finite groups”.

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Bors, A. Finite groups with an automorphism that is a complete mapping. Monatsh Math 193, 1–12 (2020). https://doi.org/10.1007/s00605-020-01395-7

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  • DOI: https://doi.org/10.1007/s00605-020-01395-7

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