Abstract
Let G be a finite group and let c(G) be the number of cyclic subgroups of G. We study the function \(\alpha (G) = c(G)/|G|\). We explore its basic properties and we point out a connection with the probability of commutation. For many families \(\mathscr {F}\) of groups we characterize the groups \(G \in \mathscr {F}\) for which \(\alpha (G)\) is maximal and we classify the groups G for which \(\alpha (G) > 3/4\). We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality \(\alpha (G) = \alpha (G/N)\) when G / N is a symmetric group.
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Acknowledgements
We would like to thank the referee for carefully reading a previous version of this paper and having very useful suggestions. Igor Lima was partially supported by bolsa de pós-doutorado 12/2014 FAPEG/CAPES.
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Garonzi, M., Lima, I. On the Number of Cyclic Subgroups of a Finite Group. Bull Braz Math Soc, New Series 49, 515–530 (2018). https://doi.org/10.1007/s00574-018-0068-x
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DOI: https://doi.org/10.1007/s00574-018-0068-x