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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.22749 |
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| _version_ | 1866917008532045824 |
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| author | Bader, Philipp |
| author_facet | Bader, Philipp |
| contents | We show that for $n \ge 6$ 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 $C(n) \subset S_n$ denote the conjugacy class of $n$-cycles and $[\cdot, \cdot]$ be the commutator of two permutations. We prove: The map $C(n) \times C(n) \to A_n, \ (τ, π) \mapsto [τ, π]$ is surjective for all $n \ge 6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Commutators of n-cycles in the symmetric group Bader, Philipp Group Theory 20B30, 20B35, 05A05 We show that for $n \ge 6$ 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 $C(n) \subset S_n$ denote the conjugacy class of $n$-cycles and $[\cdot, \cdot]$ be the commutator of two permutations. We prove: The map $C(n) \times C(n) \to A_n, \ (τ, π) \mapsto [τ, π]$ is surjective for all $n \ge 6$. |
| title | Commutators of n-cycles in the symmetric group |
| topic | Group Theory 20B30, 20B35, 05A05 |
| url | https://arxiv.org/abs/2509.22749 |