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| Format: | Preprint |
| Published: |
2020
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| Online Access: | https://arxiv.org/abs/2007.05941 |
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| _version_ | 1866911858304221184 |
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| author | Cimpoeas, Mircea |
| author_facet | Cimpoeas, Mircea |
| contents | We study the action of the groups $H(λ)$ generated by the linear fractional transformations $x:z\mapsto -\frac{1}{z} \text{ and }w:z\mapsto z+λ$, where $λ$ is a positive integer, on the subsets $\mathbb Q^*(\sqrt{n})=\{\frac{a+\sqrt n}{c}\;|\;a,b=\frac{a^2-n}{c},c\in\mathbb Z\}$, where $n$ is a square-free integer. We prove that this action has a finite number of orbits if and only if $λ=1$ or $λ=2$, and we give an upper bound for the number of orbits for $λ=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_05941 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A note on the action of Hecke groups on subsets of quadratic fields Cimpoeas, Mircea Group Theory 05A18, 05E18, 11A25, 11R11, 20F05 We study the action of the groups $H(λ)$ generated by the linear fractional transformations $x:z\mapsto -\frac{1}{z} \text{ and }w:z\mapsto z+λ$, where $λ$ is a positive integer, on the subsets $\mathbb Q^*(\sqrt{n})=\{\frac{a+\sqrt n}{c}\;|\;a,b=\frac{a^2-n}{c},c\in\mathbb Z\}$, where $n$ is a square-free integer. We prove that this action has a finite number of orbits if and only if $λ=1$ or $λ=2$, and we give an upper bound for the number of orbits for $λ=2$. |
| title | A note on the action of Hecke groups on subsets of quadratic fields |
| topic | Group Theory 05A18, 05E18, 11A25, 11R11, 20F05 |
| url | https://arxiv.org/abs/2007.05941 |