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This is a Quantitative Study study.

October 20, 2025Open Access

Commutators of n-cycles in the symmetric group

Key Points

Every even permutation on n symbols is the commutator of two n-cycles for n ≥ 6.
The map from the conjugacy class of n-cycles to the alternating group is surjective for n ≥ 6.
This result extends the understanding of commutators within symmetric groups and their structure.
The findings have implications for the study of group theory and permutation representations.

Abstract

We show that for n 6 every even permutation on n symbols is the commutator of two n-cycles. More precisely, let Sₙ be the symmetric group and Aₙ the alternating group. Let C (n) Sₙ denote the conjugacy class of n-cycles and, be the commutator of two permutations. We prove: The map C (n) C (n) Aₙ, \ (τ, π) τ, π is surjective for all n 6.

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Cite This Study

Philipp Bader (Fri,) studied this question.

synapsesocial.com/papers/68f6196ee0bbbc94fac3638a https://doi.org/https://doi.org/10.48550/arxiv.2509.22749

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