Abstract
References
- [1] M. Alaeiyan and B. Askari, Transitive permutation groups with elements of movement m or m − 1, Math. Reports, 14 (64), 317-324, 2012.
Abstract
Let $G$ be a permutation group on a set $\Omega$. Then for each $g\in G$, we define the movement of $g$, denoted by $\move(g)$, the maximal cardinality $|\Delta^{g}\backslash \Delta|$ of $\Delta^{g}\backslash \Delta$ over all subsets $\Delta$ of $\Omega$. And the movement of $G$ is defined as the maximum of $\move(g)$ over all $g\in G$, denoted by $\move(G)$. A permutation group $G$ is said to have bounded movement if it has movement bounded by some positive integer $m$, that is $\move(G)\leq m$. In this paper, we consider the finite transitive permutation groups $G$ with movement $\move(G)=m$ for some positive integer $m>4$, where $G$ is not a $2$-group but in which every non-identity element has the movement $m$ or $m-4$, and there is at least one non-identity element that has the movement $m-4$. We give a characterization for elements of $G$ in Theorem\ref{thm-1}. Further, we apply Theorem \ref{thm-1} to character transitive permutation group $G$ in Theorem \ref{thm-2}. These results give a partial answer to the open problem posed by the authors in 2024.
References
- [1] M. Alaeiyan and B. Askari, Transitive permutation groups with elements of movement m or m − 1, Math. Reports, 14 (64), 317-324, 2012.
Details
| Primary Language | English |
|---|---|
| Subjects | Group Theory and Generalisations |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | April 11, 2025 |
| Publication Date | |
| Submission Date | November 5, 2024 |
| Acceptance Date | March 23, 2025 |
| Published in Issue | Year 2025 Early Access |