Derangements in permutation groups with two orbits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915340802326528 |
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| author | Lee, Melissa Popiel, Tomasz Verret, Gabriel |
| author_facet | Lee, Melissa Popiel, Tomasz Verret, Gabriel |
| contents | A classical theorem of Jordan asserts that if a group $G$ acts transitively on a finite set of size at least $2$, then $G$ contains a derangement (a fixed-point free element). Generalisations of Jordan's theorem have been studied extensively, due in part to their applications in graph theory, number theory and topology. We address a generalisation conjectured recently by Ellis and Harper, which says that if $G$ has exactly two orbits and those orbits have equal length $n \geq 2$, then $G$ contains a derangement. We prove this conjecture in the case where $n$ is a product of two primes, and verify it computationally for $n \leq 30$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derangements in permutation groups with two orbits Lee, Melissa Popiel, Tomasz Verret, Gabriel Group Theory A classical theorem of Jordan asserts that if a group $G$ acts transitively on a finite set of size at least $2$, then $G$ contains a derangement (a fixed-point free element). Generalisations of Jordan's theorem have been studied extensively, due in part to their applications in graph theory, number theory and topology. We address a generalisation conjectured recently by Ellis and Harper, which says that if $G$ has exactly two orbits and those orbits have equal length $n \geq 2$, then $G$ contains a derangement. We prove this conjecture in the case where $n$ is a product of two primes, and verify it computationally for $n \leq 30$. |
| title | Derangements in permutation groups with two orbits |
| topic | Group Theory |
| url | https://arxiv.org/abs/2506.11396 |