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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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