Year: 2026
Type: preprint
Abstract: We show that for $n \geq 6$ each element of the commutator subgroup in the symmetric group $\mathfrak{S}_n$ resp. in the signed symmetric group $(\mathbb{Z}/2\mathbb{Z})^n\rtimes\mathfrak{S}_n$ is the commutator of two $n$-cycles resp. the commutator of two $n$-cycles with a negative sign product; with one exception. If $n \equiv 2$ $\mathrm{mod}~4$, the element $-\mathrm{id}$ of $(\mathbb{Z}/2\mathbb{Z})^n\rtimes\mathfrak{S}_n$ is not such a commutator. In the language of Coxeter groups, this yields a description of commutators of Coxeter elements in types $A$ and $B$.
Source: arXiv (Cornell University)
Authors: Bader, Philipp, Böhmler, Bernhard, Wegener, Patrick
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214043970
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