Commutators of cycles in permutation groupsVavpetič, Aleš (Avtor)
commutatorcyclepermutationalternating groupWe 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) ▫$p + q \ge 3n + 1$▫.20162016-12-02 10:41:29Neznano87582sl