Commutators of n-cycles in the symmetric group
Philipp Bader
TL;DR
The paper proves that for $n \ge 6$, every element of $A_n$ is a commutator of two $n$-cycles, i.e., the map $f: C(n) \times C(n) \to A_n$, $f(\tau,\pi)=[\tau,\pi]$, is surjective. Building on earlier work of Vavpetič and Ore, it reduces the problem to constructing, for each even conjugacy class, a permutation $\pi$ with $[\sigma_n,\pi]$ lying in that class, where $\sigma_n=(1\ 2\ \dots\ n)$ is the shift. The method introduces irreducible conjugacy classes and stitching lemmas to combine local constructions into global coverage, providing explicit $\pi$ for all irreducible types. Consequently, the surjectivity holds for all $n \ge 6$, while the analogous statement fails for $n=3,4,5$; the result strengthens Ore-type descriptions of commutators in finite symmetric groups.
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 $C(n) \times C(n) \to A_n, \ (τ, π) \mapsto [τ, π]$ is surjective for all $n \ge 6$.