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The Reidemeister spectrum of finite abelian groups

Part of: Abelian groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  06 September 2023

Pieter Senden Open the ORCID record for Pieter Senden [Opens in a new window]
Pieter Senden*
Affiliation:
Department of Mathematics, KU Leuven Kulak Kortrijk Campus, Kortrijk, Belgium ([email protected])
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Abstract

For a finite abelian group A, the Reidemeister number of an endomorphism φ is the same as the number of fixed points of φ, and the Reidemeister spectrum of A is completely determined by the Reidemeister spectra of its Sylow p-subgroups. To compute the Reidemeister spectrum of a finite abelian p-group P, we introduce a new number associated to an automorphism ψ of P that captures the number of fixed points of ψ and its (additive) multiples, we provide upper and lower bounds for that number, and we prove that every power of p between those bounds occurs as such a number.

Keywords

finite abelian groupstwisted conjugacyReidemeister numberReidemeister spectrumfixed points

MSC classification

Primary: 20K30: Automorphisms, homomorphisms, endomorphisms, etc. 20E45: Conjugacy classes
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 4 , November 2023 , pp. 940 - 959
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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