Groups having $11$ cyclic subgroups

Document Type : Research Paper

Authors

Department of Mathematics, Shiv Nadar Institution of Eminence, Dadri-201314, Gautam Buddha Nagar, India

Abstract

Let $c(G)$ denotes the number of cyclic subgroups of a finite group $G$. A group $G$ is said to be {\em $n$-cyclic}, if $c(G)=n$. In this paper, we classify all $11$-cyclic groups.

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Main Subjects


[1] A. R. Ashrafi and E. Haghi, On n-cyclic groups, Bull. Malays. Math. Sci. Soc., 42 (2019) 3233–3246.
[2] R. Belshoff, J. Dillstrom and L. Reid, Finite groups with a prescribed number of cyclic subgroups, Comm. Algebra, 47 (2019) 1043–1056.
[3] G. Frobenius, Verallgemeinerung des Sylow’schen Satzes: Über auflösbare Gruppen II, Reichsdr., 1895.
[4] The GAP Group, GAP – groups, algorithms, and programming, Version 4.12.2, 2022.
[5] M. Garonzi and I. Lima, On the number of cyclic subgroups of a finite group, Bull. Braz. Math. Soc. (N.S.), 49 (2018) 515–530.
[6] M. H. Jafari and A. R. Madadi, On the number of cyclic subgroups of a finite group, Bull. Korean Math. Soc., 54 (2017) 2141–2147.
[7] H. Kalra, Finite groups with specific number of cyclic subgroups, Proc. Indian Acad. Sci. Math. Sci., 129 (2019) 10 pp.
[8] J. B. Nganou, Converse of Lagrange’s theorem (CLT) numbers under 1000, Int. J. Group Theory, 6 (2017) 37–42.
[9] I. M. Richards, A remark on the number of cyclic subgroups of a finite group, Amer. Math. Monthly, 91 (1984) 571–572.
[10] M. Tărnăuceanu, Non-CLT groups of order pq 3 , Math. Slovaca, 64 (2014) 311–314.
[11] M. Tărnăuceanu, Finite groups with a certain number of cyclic subgroups, Amer. Math. Monthly, 122 (2015) 275–276.
[12] M. Tărnăuceanu and L. Tóth, Cyclicity degrees of finite groups, Acta Math. Hungar, 145 (2015) 489–504.
[13] L. Tóth, On the number of cyclic subgroups of a finite Abelian group, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55(103) (2012) 423–428.
[14] W. Zhou, Finite groups with small number of cyclic subgroups, arXiv preprint arXiv:1606.02431, 2016.
  • Receive Date: 28 October 2022
  • Revise Date: 29 August 2023
  • Accept Date: 26 September 2023
  • Published Online: 01 June 2024