Commutators of n-cycles in the symmetric group

2025/09/26 by Philipp Bader, Bader, Philipp
Engineering · Mathematics · #05A05 #20B30 #20B35 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Mathematics and Applications #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2509.22749

openalex publication_date 2025/09/26 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

We show that for n ≥ 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) ⊂ S_n denote the conjugacy class of n-cycles and [⋅, ⋅] be the commutator of two permutations. We prove: The map C(n) × C(n) → A_n, (τ, π) ↦ [τ, π] is surjective for all n ≥ 6.

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