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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.23390 |
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| _version_ | 1866915698198970368 |
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| author | Lucchini, Andrea Spiga, Pablo |
| author_facet | Lucchini, Andrea Spiga, Pablo |
| contents | Let $p$ be a prime number. We say that a positive integer $n$ is a Sylow $p$-number if there exists a finite group having exactly $n$ Sylow $p$-subgroups. When $p=2$, every odd integer is a Sylow $2$-number. In contrast, when $p$ is odd, there exist two positive constants $c_p$ and $c_p^\prime$ such that, denoting by $β(p,x)$ the number of Sylow $p$-numbers less than or equal to $x$,
\[c_p\,x(\log x)^{\frac{1}{p-1}-1}
\leq β(p,x)\leq
c_p^\prime\,x(\log x)^{\frac{1}{p-1}-1}.
\]
Moreover if $β_s(p,x)$ is the number of positive integers $n\le x$ such that $n$ is the Sylow $p$-number of some finite solvable group then $$β_s(p,x)\sim c_p\,x(\log x)^{\,\frac{1}{p-1}-1}
\qquad\text{as } x\to\infty.$$
In particular, when $p$ is odd, the natural density of Sylow $p$-numbers is $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23390 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the density of Sylow numbers Lucchini, Andrea Spiga, Pablo Group Theory 20D20 Let $p$ be a prime number. We say that a positive integer $n$ is a Sylow $p$-number if there exists a finite group having exactly $n$ Sylow $p$-subgroups. When $p=2$, every odd integer is a Sylow $2$-number. In contrast, when $p$ is odd, there exist two positive constants $c_p$ and $c_p^\prime$ such that, denoting by $β(p,x)$ the number of Sylow $p$-numbers less than or equal to $x$, \[c_p\,x(\log x)^{\frac{1}{p-1}-1} \leq β(p,x)\leq c_p^\prime\,x(\log x)^{\frac{1}{p-1}-1}. \] Moreover if $β_s(p,x)$ is the number of positive integers $n\le x$ such that $n$ is the Sylow $p$-number of some finite solvable group then $$β_s(p,x)\sim c_p\,x(\log x)^{\,\frac{1}{p-1}-1} \qquad\text{as } x\to\infty.$$ In particular, when $p$ is odd, the natural density of Sylow $p$-numbers is $0$. |
| title | On the density of Sylow numbers |
| topic | Group Theory 20D20 |
| url | https://arxiv.org/abs/2512.23390 |