Vertex-primitive s-arc-transitive digraphs of symplectic groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914920651554816 |
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| author | Chen, Lei Giudici, Michael Praeger, Cheryl E. |
| author_facet | Chen, Lei Giudici, Michael Praeger, Cheryl E. |
| contents | A digraph is $s$-arc-transitive if its automorphism group is transitive on directed paths with $s$ edges, that is, on $s$-arcs. Although infinite families of finite $s$-arc transitive digraphs of arbitrary valency were constructed by the third author in 1989, existence of a vertex-primitive $2$-arc-transitive digraph was not known until an infinite family was constructed by the second author with Li and Xia in 2017. This led to a conjecture by the second author and Xia in 2018 that, for a finite vertex-primitive $s$-arc-transitive digraph, $s$ is at most $2$, together with their proof that it is sufficient to prove the conjecture for digraphs with an almost simple group of automorphisms. This paper confirms the conjecture for finite symplectic groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vertex-primitive s-arc-transitive digraphs of symplectic groups Chen, Lei Giudici, Michael Praeger, Cheryl E. Combinatorics Group Theory A digraph is $s$-arc-transitive if its automorphism group is transitive on directed paths with $s$ edges, that is, on $s$-arcs. Although infinite families of finite $s$-arc transitive digraphs of arbitrary valency were constructed by the third author in 1989, existence of a vertex-primitive $2$-arc-transitive digraph was not known until an infinite family was constructed by the second author with Li and Xia in 2017. This led to a conjecture by the second author and Xia in 2018 that, for a finite vertex-primitive $s$-arc-transitive digraph, $s$ is at most $2$, together with their proof that it is sufficient to prove the conjecture for digraphs with an almost simple group of automorphisms. This paper confirms the conjecture for finite symplectic groups. |
| title | Vertex-primitive s-arc-transitive digraphs of symplectic groups |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2408.12074 |