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Commutators of cycles in permutation groups
Vavpetič, Aleš (Author)

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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) ▫$p + q \ge 3n + 1$▫.

Language:English
Keywords:commutator, cycle, permutation, alternating group
Tipology:1.01 - Original Scientific Article
Organization:FMF - Faculty of Mathematics and Physics
Year:2016
Number of pages:str. 67-77
Numbering:Vol. 10, no. 1
UDC:512.542
ISSN on article:1855-3966
COBISS.SI-ID:17731929 Link is opened in a new window
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Record is a part of a journal

Title:Ars mathematica contemporanea
Publisher:Društvo matematikov, fizikov in astronomov
ISSN:1855-3966
COBISS.SI-ID:239049984 This link opens in a new window

Secondary language

Language:Slovenian
Title:Komutatorji ciklov v permutacijskih grupah
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

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