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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2411.02219 |
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| _version_ | 1866917828068638720 |
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| author | Jones, Gareth A. |
| author_facet | Jones, Gareth A. |
| contents | We obtain formulae for the numbers of isomorphism and conjugacy classes of non-identity proper subgroups of the groups $G={\rm PSL}_2(p)$, $p$ prime, and for the numbers of those conjugacy classes which do or do not consist of self-normalising subgroups. The formulae are used to prove lower bounds $17$, $18$, $6$ and $12$ respectively satisfied by these invariants for all $p>37$. A computer search carried out for a different problem shows that these bounds are attained for over a million primes $p$; we show that if the Bateman--Horn Conjecture is true, they are attained for infinitely many primes. Also, assuming no unproved conjectures, we use a result of Heath-Brown to obtain upper bounds for these invariants, valid for an infinite set of primes $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting conjugacy classes of subgroups of ${\rm PSL}_2(p)$ Jones, Gareth A. Group Theory Number Theory Primary 20D99, secondary 11N32, 20D60, 20E32, 20G40 We obtain formulae for the numbers of isomorphism and conjugacy classes of non-identity proper subgroups of the groups $G={\rm PSL}_2(p)$, $p$ prime, and for the numbers of those conjugacy classes which do or do not consist of self-normalising subgroups. The formulae are used to prove lower bounds $17$, $18$, $6$ and $12$ respectively satisfied by these invariants for all $p>37$. A computer search carried out for a different problem shows that these bounds are attained for over a million primes $p$; we show that if the Bateman--Horn Conjecture is true, they are attained for infinitely many primes. Also, assuming no unproved conjectures, we use a result of Heath-Brown to obtain upper bounds for these invariants, valid for an infinite set of primes $p$. |
| title | Counting conjugacy classes of subgroups of ${\rm PSL}_2(p)$ |
| topic | Group Theory Number Theory Primary 20D99, secondary 11N32, 20D60, 20E32, 20G40 |
| url | https://arxiv.org/abs/2411.02219 |