Inequalities detecting structural properties of a finite group. (English) Zbl 1388.20039
Summary: We prove several results detecting cyclicity or nilpotency of a finite group \(G\) in terms of inequalities involving the orders of the elements of \(G\) and the orders of the elements of the cyclic group of order \(|G|\). We prove that, among the groups of the same order, the number of cyclic subgroups is minimal for the cyclic group, and the product of the orders of the elements is maximal for the cyclic group.
MSC:
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20D30 | Series and lattices of subgroups |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 05A20 | Combinatorial inequalities |
Software:
MathOverflowOnline Encyclopedia of Integer Sequences:
Possible values for the product of the orders of all elements of a finite group.References:
| [1] | DOI: 10.1080/00927870802502530 · Zbl 1183.20022 · doi:10.1080/00927870802502530 |
| [2] | De Medts, T. (2012). Order increasing bijection from arbitrary groups to cyclic groups. Available at: http://mathoverflow.net/questions/104183/. |
| [3] | De Medts T., Bull. Belg. Math. Soc. Simon Stevin 15 pp 699– (2008) |
| [4] | De Medts, T., Tarnauceanu, M. (2012). An inequality detecting nilpotency of finite groups. Available at: http://arxiv.org/abs/1207.1020. |
| [5] | Frobenius, F. G. (1907). Über einen Fundamentalsatz der Gruppentheorie II, Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin), 428–437. · JFM 38.0174.01 |
| [6] | DOI: 10.2307/2324902 · doi:10.2307/2324902 |
| [7] | DOI: 10.2307/2695368 · doi:10.2307/2695368 |
| [8] | Tarnauceanu, M. (2011). A question on the product of element orders of a finite group. Available at: http://mathoverow.net/questions/82547/. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
