Commutators of cycles in permutation groups

Vavpetič, Aleš (author)

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
• Typology: 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
• PID: 20.500.12556/RUL-87582
• UDC: 512.542
• ISSN on article: 1855-3966
• COBISS.SI-ID: 17731929
• Publication date in RUL: 02.12.2016

Record is a part of a journal

• Title: Ars mathematica contemporanea
• Publisher: Društvo matematikov, fizikov in astronomov, Društvo matematikov, fizikov in astronomov, Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije
• ISSN: 1855-3966
• COBISS.SI-ID: 239049984

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