Hostname: page-component-6bf8c574d5-pdxrj Total loading time: 0 Render date: 2025-03-03T21:28:23.920Z Has data issue: false hasContentIssue false
  • English
  • Français

The Reidemeister spectrum of finite abelian groups

Part of: Structure and classification of infinite or finite groups Abelian 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 (pieter.senden@kuleuven.be)
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bleak, C., Fel’shtyn, A. L. and Gonçalves, D. L., Twisted conjugacy classes in R. Thompson’s group F, Pacific J. Math. 238(1) (2008), 1–6. doi: 10.2140/pjm.2008.238.1.CrossRefGoogle Scholar
Dekimpe, K. and Gonçalves, D. L., The $R_\infty$ property for free groups, free nilpotent groups and free solvable groups, Bull. Lond. Math. Soc. 46(4) (2014), 737–746. doi: 10.1112/blms/bdu029.CrossRefGoogle Scholar
Dekimpe, K. and Gonçalves, D. L., The $R_\infty$ property for abelian groups, Topol. Methods Nonlinear Anal. 46(2) (2015), 773–784. doi: 10.12775/TMNA.2015.066.Google Scholar
Dekimpe, K., Kaiser, T. and Tertooy, S., The Reidemeister spectra of low dimensional crystallographic groups, J. Algebra 533 (2019), 353–375. doi: 10.1016/j.jalgebra.2019.04.038.CrossRefGoogle Scholar
Dekimpe, K., Tertooy, S. and Vargas, A. R., Fixed points of diffeomorphisms on nilmanifolds with a free nilpotent fundamental group, Asian J. Math. 24(1) (2020), 147–164. doi: 10.4310/AJM.2020.v24.n1.a6.CrossRefGoogle Scholar
Fel’shtyn, A. L. and Gonçalves, D. L., Twisted conjugacy classes of automorphisms of Baumslag-Solitar groups, Algebra Discrete Math. 5(3) (2006), 36–48.Google Scholar
Fel’shtyn, A. L. and Hill, R., The Reidemeister zeta function with applications to Nielsen Theory and a connection with Reidemeister torsion, K-Theory 8(4) (1994), 367–393. doi: 10.1007/BF00961408.CrossRefGoogle Scholar
Fel’shtyn, A. L. and Nasybullov, T., The $R_\infty$ and $S_\infty$ properties for linear algebraic groups, J. Group Theory 19(5) (2016), 901–921. doi: 10.1515/jgth-2016-0004.CrossRefGoogle Scholar
Goldsmith, B., Karimi, F. and White, N., On the Reidemeister spectrum of an Abelian group, Forum Math. 31(1) (2019), 199–214. doi: 10.1515/forum-2017-0184.CrossRefGoogle Scholar
Gonçalves, D. L. and Wong, P. N., Twisted conjugacy classes in nilpotent groups, J. für die Reine und Angew. Math. 2009(633) (2009), 11–27. doi: 10.1515/CRELLE.2009.058.Google Scholar
Hillar, C. J. and Rhea, D. L., Automorphisms of finite abelian groups, Amer. Math. Monthly 114(10) (2007), 917–923. doi: 10.1080/00029890.2007.11920485.CrossRefGoogle Scholar
Kerby, B. and Rode, E., Characteristic subgroups of finite abelian groups, Comm. Algebra 39(4) (2011), 1315–1343. doi: 10.1080/00927871003591843.CrossRefGoogle Scholar
Mubeena, T. and Sankaran, P., Twisted conjugacy classes in abelian extensions of certain linear groups, Canad. Math. Bull. 57(1) (2014), 132–140. doi: 10.4153/CMB-2012-013-7.CrossRefGoogle Scholar
Roman’kov, V., Twisted conjugacy classes in nilpotent groups, J. Pure Appl. Algebra 215(4) (2011), 664–671. doi: 10.1016/j.jpaa.2010.06.015.CrossRefGoogle Scholar
Senden, P., How does the structure of a group determine its Reidemeister spectrum?. PhD thesis, KU Leuven, 2023.Google Scholar
Senden, P., The Reidemeister spectrum of split metacyclic groups, arXiv:2109.12892 [math.GR], 2021. doi: 10.48550/arXiv.2109.12892.CrossRefGoogle Scholar
Senden, P., Twisted conjugacy in direct products of groups, Comm. Algebra 49(12) (2021), 5402–5422. doi: 10.1080/00927872.2021.1945615.CrossRefGoogle Scholar
Taback, J. and Wong, P. N., A note on twisted conjugacy classes and generalized Baumslag-Solitar groups, 2008. arXiv: 0606284v3 [math.GR]. doi: 10.48550/arXiv.math/0606284.CrossRefGoogle Scholar
Tertooy, S., TwistedConjugacy, Computation with twisted conjugacy classes, Version 2.0.0. https://sTertooy.github.io/TwistedConjugacy/. GAP package. 2021.Google Scholar