Commutators of n-cycles in the symmetric group

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This is a Quantitative Study study.

Philipp Bader

doi:10.48550/arxiv.2509.22749

September 26, 2025Open Access

Commutators of n-cycles in the symmetric group

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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.

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Implication

This work shows every even permutation can be expressed as commutators of n-cycles, indicating progress in group theory.

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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Philipp Bader (2025) studied this question.

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

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Commutators of n-cycles in the symmetric group

Key Points

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Implication

Abstract

Discussion

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Commutators of n-cycles in the symmetric group

Key Points

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Implication

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Discussion

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