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On Conjugacy Classes of Derangements in Symmetric and Alternating Groups — arXiv:2608.22428

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.22428
License (as archived): CC0-1.0
Archived: 2026-10-18T08:35:56.657Z
SHA-256 of the archived text: 34cde6498e5c8fefc2449b1bec5b0b209750ea4077c5b835dc4367e75340b034
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How to cite

On Conjugacy Classes of Derangements in Symmetric and Alternating Groups — arXiv:2608.22428 — arXiv (en), https://arxiv.org/abs/2608.22428 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_fb0da355d4e4 (Metraipolis archive dc_fb0da355d4e4, sha256 34cde6498e5c8fefc2449b1bec5b0b209750ea4077c5b835dc4367e75340b034)

BibTeX

@misc{dcfb0da355d4e4,
  title = {On Conjugacy Classes of Derangements in Symmetric and Alternating Groups — arXiv:2608.22428},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.22428},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_fb0da355d4e4; sha256 34cde6498e5c8fefc2449b1bec5b0b209750ea4077c5b835dc4367e75340b034},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.22428. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In this article, we prove two conjectures of Burness and Fusari [Timothy Burness and Marco Fusari, On derangements in simple permutation groups, Forum Math. Sigma 13 (2025)] concerning the powers and products of conjugacy classes of derangements in the symmetric and alternating groups: (1) We show that there exist two conjugacy classes $C$ and $D$ of derangements in $S_n$ such that $S_n=C^2\cup CD$, and (2) We show that there exists a conjugacy class $C$ of derangements in $A_n$ such that $C^2=A_n$, whenever $n\equiv 3\;(\text{mod}\;4)$. In fact, our result concerning the second conjecture holds in a considerably more general setting, which also answers affirmatively a question posed by Bertram [Edward Bertram, Even permutations as a product of two conjugate cycles, J. Comb. Theory, Ser. A 12 (1972), 368-380] in a particular case. Moreover, we show that any conjugacy class $C$ of derangements in $S_n$ (resp. $A_n$) contains a pair of elements that generate $S_n$ or $A_n$ (resp. $A_n$), unless $C$ is the conjugacy class of fixed-point-free involutions.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-25 in math.CO).
Categorie arXiv: Group Theory (math.GR) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2608.22428 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-18T08:27:54.474Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-25?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).