# zbMATH — the first resource for mathematics

The second minimum/maximum value of the number of cyclic subgroups of finite $$p$$-groups. (English) Zbl 07302540
Summary: Let $$C(G)$$ be the poset of cyclic subgroups of a finite group $$G$$ and let $$\mathscr{P}$$ be the class of $$p$$-groups of order $$p^n (n \geq 3)$$. Consider the function $$\alpha : \mathscr{P} \longrightarrow (0, 1]$$ given by $$\alpha (G) = |C(G)| / |G|$$. In this paper, we determine the second minimum value of $$\alpha$$, as well as the corresponding minimum points. Since the problem of finding the second maximum value of $$\alpha$$ has been solved for $$p = 2$$, we focus on the case of odd primes in determining the second maximum.
##### MSC:
 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D15 Finite nilpotent groups, $$p$$-groups 20D25 Special subgroups (Frattini, Fitting, etc.)
GAP
Full Text:
##### References:
 [1] Aivazidis, S. and Müller, T., ‘Finite non-cyclic p-groups whose number of subgroups is minimal’, Arch. Math. (Basel)114(1) (2020), 13-17. · Zbl 1445.20014 [2] Berkovich, Y., Groups of Prime Power Order, Vol. 1 (de Gruyter, Berlin, 2008). · Zbl 1168.20001 [3] Burnside, W., Theory of Groups of Finite Order, 2nd edn (Dover Publications, Inc., New York, 1955). · Zbl 0064.25105 [4] Butler, L. M., ‘Subgroup lattices and symmetric functions’, Mem. Amer. Math. Soc.112 (1994), 1-160. · Zbl 0813.05067 [5] Garonzi, M. and Lima, I., ‘On the number of cyclic subgroups of a finite group’, Bull. Braz. Math. Soc.49 (2018), 515-530. · Zbl 06971581 [6] Jafari, M. H. and Madadi, A. R., ‘On the number of cyclic subgroups of a finite group’, Bull. Korean Math. Soc.54(60) (2017), 2141-2147. · Zbl 1432.20018 [7] Laffey, T. J., ‘The number of solutions of x^3 = 1 in a 3-group’, Math. Z.149(2) (1976), 43-45. · Zbl 0314.20020 [8] Lazorec, M. S., ‘A connection between the number of subgroups and the order of a finite group’, Preprint, 2019, arXiv:1901.06425. [9] Lazorec, M. S. and Tărnăuceanu, M., ‘A note on the number of cyclic subgroups of a finite group’, Bull. Math. Soc. Sci. Math. Roumanie62/110(4) (2019), 403-416. [10] Miller, G. A., ‘An extension of Sylow’s theorem’, Proc. Lond. Math. Soc. Ser. 22 (1905), 142-143. [11] Miller, G. A., ‘The groups of order p^m which contain exactly p cyclic subgroups of order p^a’, Trans. Amer. Math. Soc.7(2) (1906), 228-232. · JFM 37.0169.01 [12] Qu, H., ‘Finite non-elementary abelian p-groups whose number of subgroups is maximal’, Israel J. Math.195 (2013), 773-781. · Zbl 1285.20017 [13] Richards, I. M., ‘A remark on the number of cyclic subgroups of a finite group’, Amer. Math. Monthly91(9) (1984), 571-572. [14] Sædén Ståhl, G., Laine, J. and Behm, G., On $$p$$-groups of Low Power Order, Bachelor Thesis, KTH Royal Institute of Technology, 2010. [15] Suzuki, M., Group Theory, Vols. I, II (Springer, Berlin, 1982, 1986). [16] Tărnăuceanu, M., ‘On a conjecture by Haipeng Qu’, J. Group Theory22(3) (2019), 505-514. · Zbl 1441.20012 [17] Tărnăuceanu, M. and Tóth, L., ‘Cyclicity degrees of finite groups’, Acta Math. Hungar.145 (2015), 489-504. · Zbl 1348.20027 [18] , GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018, https://www.gap-system.org.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.