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On the Number of Cyclic Subgroups of a Finite Group

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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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References

  • Brauer, R., Fowler, K.A.: On groups of even order. Ann. Math. 2(62), 565–583 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  • Doerk, K., Hawkes, T.: Finite Soluble Groups, de Gruyter Expositions in Mathematics 4, de Gruyter (1992)

  • Garonzi, M., Patassini, M.: Inequalities detecting structural properties of a finite group. Commun. Algebra 45(2), 677–687 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Guralnick, R.M., Robinson, G.R.: On the commuting probability in finite groups. J. Algebra 300(2), 509–528 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Knorr, R.: On the number of characters in a \(p\)-block of a \(p\)-solvable group. Ill. J. Math. 28, 181–210 (1984)

    MathSciNet  MATH  Google Scholar 

  • Leemans, D., Vauthier, L.: An atlas of abstract regular polytopes for small groups. Aequationes Math. 72(3), 313–320 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, D.J.S.: A Course in the Theory of Groups; Second edition; Graduate Texts in Mathematics, 80. Springer, New York (1996)

    Book  Google Scholar 

  • Schur, J: On the representation of the symmetric and alternating groups by fractional linear substitutions. Translated from the German[J. Reine Angew. Math. 139 (1911), 155–250] by Marc-Felix Otto. Int. J. Theoret. Phys. 40(1):413–458 (2001)

  • Tarnauceanu, M.: Finite groups with a certain number of cyclic subgroups. Am. Math. Mon. 122(3), 275–276 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.10. https://www.gap-system.org (2018)

  • Wall, C.T.C.: On groups consisting mostly of involutions. Proc. Camb. Philos. Soc. 67, 251–262 (1970)

    Article  MathSciNet  MATH  Google Scholar 

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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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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, Campus Universitário Darcy Ribeiro, Brasília, DF, 70910-900, Brazil

    Martino Garonzi

  2. Universidade Federal de Goiás, IMTec - Regional Catalão, Av. Dr. Lamartine P. de Avelar, 1120 Setor Universitário, Catalão, GO, 75704-020, Brazil

    Igor Lima

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  1. Martino Garonzi
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  2. Igor Lima
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Correspondence to Martino Garonzi.

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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

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