Products of three conjugacy classes in the alternating group
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909605988139008 |
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| author | Dona, Daniele |
| author_facet | Dona, Daniele |
| contents | We prove that for $δ$ small, $n$ large, and any three conjugacy classes $C_{1},C_{2},C_{3}$ of $G=\mathrm{Alt}(n)$ of size at least $|G|^{1-δ}$ we have $C_{1}C_{2}C_{3}=G$.
The result provides a positive answer to Problem 20.23 of the Kourovka Notebook [KM22], improves theorems of Garonzi and Maróti [GM21] (using $4$ classes) and Rodgers [Rod02] (using larger classes), complements the known result for $G$ a simple group of Lie type [MP21] [LST24] [FM25], and is tight in several senses. Furthermore, since no character theory is involved, the proof can be used in principle to build a constructive algorithm that, given $g\in G$, outputs $c_{i}\in C_{i}$ such that $c_{1}c_{2}c_{3}=g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Products of three conjugacy classes in the alternating group Dona, Daniele Group Theory Combinatorics 20B30, 20E45 We prove that for $δ$ small, $n$ large, and any three conjugacy classes $C_{1},C_{2},C_{3}$ of $G=\mathrm{Alt}(n)$ of size at least $|G|^{1-δ}$ we have $C_{1}C_{2}C_{3}=G$. The result provides a positive answer to Problem 20.23 of the Kourovka Notebook [KM22], improves theorems of Garonzi and Maróti [GM21] (using $4$ classes) and Rodgers [Rod02] (using larger classes), complements the known result for $G$ a simple group of Lie type [MP21] [LST24] [FM25], and is tight in several senses. Furthermore, since no character theory is involved, the proof can be used in principle to build a constructive algorithm that, given $g\in G$, outputs $c_{i}\in C_{i}$ such that $c_{1}c_{2}c_{3}=g$. |
| title | Products of three conjugacy classes in the alternating group |
| topic | Group Theory Combinatorics 20B30, 20E45 |
| url | https://arxiv.org/abs/2505.06012 |