Commutators of n-cycles in the symmetric group

Philipp Bader

Commutators of n-cycles in the symmetric group

Group Theory 2025-10-14 v2

Authors: Philipp Bader

Abstract

We show that for $n \ge 6$ every even permutation on $n$ symbols is the commutator of two $n$ -cycles. More precisely, let $S_n$ be the symmetric group and $A_n$ the alternating group. Let $C(n) \subset S_n$ denote the conjugacy class of $n$ -cycles and $[\cdot, \cdot]$ be the commutator of two permutations. We prove: The map $C(n) \times C(n) \to A_n, \ (\tau, \pi) \mapsto [\tau, \pi]$ is surjective for all $n \ge 6$ .

Keywords

symmetric group alternating group n-cycles commutators conjugacy class surjectivity

Cite

@article{arxiv.2509.22749,
  title  = {Commutators of n-cycles in the symmetric group},
  author = {Philipp Bader},
  journal= {arXiv preprint arXiv:2509.22749},
  year   = {2025}
}

Comments

The result presented in this article can be derived from [Ale\v{s} Vavpeti\v{c}, Commutators of cycles in permutation groups, Ars Math. Contemp. 10 (2016)] and is therefore not intended for publication. We thank Goulnara Arzhantseva and Ale\v{s} Vavpeti\v{c} for pointing out the reference as well as for the subsequent discussion and explanations

Commutators of n-cycles in the symmetric group

Abstract

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