On Groups of Linear Fractional Transformations Stabilizing Finite Sets of Four Elements
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910991334244352 |
|---|---|
| author | Fonga, Patrick Nyadjo |
| author_facet | Fonga, Patrick Nyadjo |
| contents | Let $E$ be a subset of the projective line over a commutative field $\mathbb{K}$. When $\mathbb{K}$ has infinite cardinality, it is well known that if $E$ contains at most three elements, then the group of linear fractional transformations preserving $E$ is either infinite or isomorphic to the symmetric group on three elements. In this work, we investigate the case where $E$ consists of four elements. We show that the group of projective linear transformations stabilizing $E$ is, depending on the characteristic of the field $\mathbb{K}$, isomorphic to either the Klein four-group $V_4$, the dihedral group $D_4$ of order eight, the alternating group $\mathfrak{A}_4$ of order twelve, or the symmetric group $\mathfrak{S}_4$ of order twenty-four. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Groups of Linear Fractional Transformations Stabilizing Finite Sets of Four Elements Fonga, Patrick Nyadjo Number Theory Group Theory Let $E$ be a subset of the projective line over a commutative field $\mathbb{K}$. When $\mathbb{K}$ has infinite cardinality, it is well known that if $E$ contains at most three elements, then the group of linear fractional transformations preserving $E$ is either infinite or isomorphic to the symmetric group on three elements. In this work, we investigate the case where $E$ consists of four elements. We show that the group of projective linear transformations stabilizing $E$ is, depending on the characteristic of the field $\mathbb{K}$, isomorphic to either the Klein four-group $V_4$, the dihedral group $D_4$ of order eight, the alternating group $\mathfrak{A}_4$ of order twelve, or the symmetric group $\mathfrak{S}_4$ of order twenty-four. |
| title | On Groups of Linear Fractional Transformations Stabilizing Finite Sets of Four Elements |
| topic | Number Theory Group Theory |
| url | https://arxiv.org/abs/2506.00729 |