Subgroups of symmetric groups: enumeration and asymptotic properties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910863633416192 |
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| author | Roney-Dougal, Colva M. Tracey, Gareth |
| author_facet | Roney-Dougal, Colva M. Tracey, Gareth |
| contents | In this paper, we prove that the symmetric group $\mathrm{S}_n$ has $2^{n^2/16+o(n^2)}$ subgroups, settling a conjecture of Pyber from 1993. We also derive asymptotically sharp upper and lower bounds on the number of subgroups of $\mathrm{S}_n$ of various kinds, including the number of $p$-subgroups. In addition, we prove a range of theorems about random subgroups of $\mathrm{S}_n$. In particular, we prove the surprising result that for infinitely many $n$, the probability that a random subgroup of $\mathrm{S}_n$ is nilpotent is bounded away from $1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_05416 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subgroups of symmetric groups: enumeration and asymptotic properties Roney-Dougal, Colva M. Tracey, Gareth Group Theory 20B35, 20F69, 05A16, 20E07, 20E25 In this paper, we prove that the symmetric group $\mathrm{S}_n$ has $2^{n^2/16+o(n^2)}$ subgroups, settling a conjecture of Pyber from 1993. We also derive asymptotically sharp upper and lower bounds on the number of subgroups of $\mathrm{S}_n$ of various kinds, including the number of $p$-subgroups. In addition, we prove a range of theorems about random subgroups of $\mathrm{S}_n$. In particular, we prove the surprising result that for infinitely many $n$, the probability that a random subgroup of $\mathrm{S}_n$ is nilpotent is bounded away from $1$. |
| title | Subgroups of symmetric groups: enumeration and asymptotic properties |
| topic | Group Theory 20B35, 20F69, 05A16, 20E07, 20E25 |
| url | https://arxiv.org/abs/2503.05416 |