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October 20, 20250 citationsOpen Access

Commutators of n-Cycles in the Symmetric Group for n ≥ 6

Commutators of n-cycles in the symmetric group

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PBPhilipp Bader

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Overview

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

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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Cite This Study

Philipp Bader (2025) studied this question.

synapsesocial.com/papers/68f6196ee0bbbc94fac3638ahttps://doi.org/10.48550/arxiv.2509.22749
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Also Consider

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  1. 1$GL_n(\mathbb{F}_q)$-analogues of some properties of $n$-cycles in $\mathfrak{S}_n$2024
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  5. 5On the number of cycles in commutators of random permutations2024