Transitive sets of derangements in primitive actions of PSL_2(q)
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917163890114560 |
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| author | Müller, Peter |
| author_facet | Müller, Peter |
| contents | Problem 8.75 of the Kourovka Notebook [10], attributed to John G. Thompson, asks the following: Suppose $G$ is a finite primitive permutation group on $Ω$, and $α$, $β$ are distinct points of $Ω$. Does there exist an element $g\in G$ such that $α^g=β$ and $g$ fixes no point of $Ω$? A recent negative example is given in [12], where $G$ is the Steinberg triality group ${}^{3}D_{4}(2)$ acting primitively on 4,064,256 points. At present this is the only negative example known. In this note we show that almost simple primitive permutation groups with socle isomorphic to PSL_2(q) do not give negative examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transitive sets of derangements in primitive actions of PSL_2(q) Müller, Peter Group Theory 20B15 (Primary) Problem 8.75 of the Kourovka Notebook [10], attributed to John G. Thompson, asks the following: Suppose $G$ is a finite primitive permutation group on $Ω$, and $α$, $β$ are distinct points of $Ω$. Does there exist an element $g\in G$ such that $α^g=β$ and $g$ fixes no point of $Ω$? A recent negative example is given in [12], where $G$ is the Steinberg triality group ${}^{3}D_{4}(2)$ acting primitively on 4,064,256 points. At present this is the only negative example known. In this note we show that almost simple primitive permutation groups with socle isomorphic to PSL_2(q) do not give negative examples. |
| title | Transitive sets of derangements in primitive actions of PSL_2(q) |
| topic | Group Theory 20B15 (Primary) |
| url | https://arxiv.org/abs/2512.19500 |