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On the possible number of elements of given order in a finite group

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Abstract

The main motivation of this paper is to introduce a problem of some combinatorial flavor about finite groups which seems to be new in the literature. Letk>1 be a fixed positive integer and denote byf(k, G) the number of elements of orderk in the groupG. We examine the setF(k)={f(k, G)| G a finite group}/{0}. We give a complete characterization ofF(k) if 4|k ork=6 and show some modest partial results for certain other values ofk. It seems to us that the question is surprisingly difficult even in such simple cases ask=3, which we investigate in detail.

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Authors and Affiliations

  1. Department of Algebra and Number Theory, Eötvös University, Múzeum krt. 6-8, H-1088, Budapest, Hungary

    Róbert Freud

  2. Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, H-1364, Budapest, Hungary

    Péter Pál Pálfy

Authors
  1. Róbert Freud
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  2. Péter Pál Pálfy
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Corresponding author

Correspondence to Róbert Freud.

Additional information

Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1901.

Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA), Grant No. 1903.

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Freud, R., Pálfy, P.P. On the possible number of elements of given order in a finite group. Israel J. Math. 93, 345–358 (1996). https://doi.org/10.1007/BF02761111

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  • DOI: https://doi.org/10.1007/BF02761111

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