Commutators of cycles in permutation groups

Abstract

Dokažemo, da je za ▫$n \ge 5$▫ vsak element alternirajoče grupe ▫$A_n$▫ komutator dveh ciklov ▫$A_n$▫. Dokažemo tudi, da je za ▫$n \ge 2$▫ vsak ▫$(2n + 1)$▫-cikel permutacijske grupe ▫$S_{2n + 1}$▫ komutator ▫$p$▫-cikla in ▫$q$▫-cikla ▫$S_{2n + 1}$▫, če in samo če so izpolnjeni naslednji trije pogoji: (i)▫ $n + 1 \le p, q$▫, (ii) ▫$2n + 1 \ge p, q$▫, (iii) ▫$p + q \ge 3n + 1$▫.

Keywords

komutator;cikel;permutacija;alternirajoča grupa;commutator;cycle;permutation;alternating group;

Data

Language:	English
Year of publishing:	2016
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	512.542
COBISS:	17731929
ISSN:	1855-3966
Parent publication:	Ars mathematica contemporanea
Views:	979
Downloads:	0
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Komutatorji ciklov v permutacijskih grupah
Secondary abstract:	We prove that for ▫$n \ge 5$▫, every element of the alternating group ▫$A_n$▫ is a commutator of two cycles of ▫$A_n$▫. Moreover we prove that for ▫$n \ge 2$▫, a ▫$(2n + 1)$▫-cycle of the permutation group ▫$S_{2n + 1}$▫ is a commutator of a ▫$p$▫-cycle and a ▫$q$▫-cycle of ▫$S_{2n + 1}$▫ if and only if the following three conditions are satisfied: (i) ▫$n + 1 \le p, q$▫, (ii) ▫$2n + 1 \ge p, q$▫, (iii) ▫$p + q \ge 3n + 1$▫.
Secondary keywords:	komutator;cikel;permutacija;alternirajoča grupa;
Pages:	str. 67-77
Volume:	ǂVol. ǂ10
Issue:	ǂno. ǂ1
Chronology:	2016
ID:	9233736