If for each subset $\Gamma$ of $\Omega$ the size $|\Gamma^g\setminus\Gamma|$ is bounded, for $g\in G,$ we define the movement of $g$ as the $\max|\Gamma^g\setminus\Gamma|$ over all subsets $\Gamma$ of $\Omega,$ and the movement of $G$ is defined as the maximum of move$(g)$ over all non-identity elements of $g\in G.$ In this paper we classify all transitive permutation groups with bounded movement equal to $m$ that are not a $2$-group, but in which every non-identity element has movement $m$ or $m-2$.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $G$ be a permutation group on a set $\Omega$ with no fixed points in $\Omega$ and let $m$ be a positive integer.
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Key Findings & Data:Let $G$ be a permutation group on a set $\Omega$ with no fixed points in $\Omega$ and let $m$ be a positive integer.
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Conclusion & Impact:If for each subset $\Gamma$ of $\Omega$ the size $|\Gamma^g\setminus\Gamma|$ is bounded, for $g\in G,$ we define the movement of $g$ as the $\max|\Gamma^g\setminus\Gamma|$ over all subsets $\Gamma$ of $\Omega,$ and the movement of $G$ is defined as the maximum of move$(g)$ over all non-identity elements of $g\in G.$ In this paper we classify all transitive permutation groups with bounded movement equal to $m$ that are not a $2$-group, but in which every non-identity element has movement $m$ or $m-2$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:permutation group · transitive · movement
Abstract
Let $G$ be a permutation group on a set $\Omega$ with no fixed points in $\Omega$ and let $m$ be a positive integer. If for each subset $\Gamma$ of $\Omega$ the size $|\Gamma^g\setminus\Gamma|$ is bounded, for $g\in G,$ we define the movement of $g$ as the $\max|\Gamma^g\setminus\Gamma|$ over all subsets $\Gamma$ of $\Omega,$ and the movement of $G$ is defined as the maximum of move$(g)$ over all non-identity elements of $g\in G.$ In this paper we classify all transitive permutation groups with bounded movement equal to $m$ that are not a $2$-group, but in which every non-identity element has movement $m$ or $m-2$.
Alaeiyan, M., Shabeeb, M., Akbarizadeh, M. (2024). Transitive permutation groups with elements of movement $m$ or $m-2$. Hacettepe Journal of Mathematics and Statistics, Volume: 53 Issue: 4. https://doi.org/10.15672/hujms.1223815
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M. Alaeiyan and B. Askari, Transitive permutation groups with elements of movement m or m-1, Math. Reports 14 (64), 4, 317-324, 2012.