On transitive permutation groups with bounded movement

Year 2025, Volume: 54 Issue: 6, 2278 - 2294, 30.12.2025

Hailin Liu , Longzhi Lu

https://doi.org/10.15672/hujms.1579866

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 ${\rm move}(g)$, 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 ${\rm move}(g)$ over all $g\in G$, denoted by ${\rm move}(G)$. A permutation group $G$ is said to have bounded movement if it has movement bounded by some positive integer $m$, that is ${\rm move}(G)\leq m$. In this paper, we consider the finite transitive permutation groups $G$ with movement ${\rm move}(G)=m$ 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 1.1. Further, we apply Theorem 1.1 to characterize transitive permutation group $G$ in Theorem 1.2. These results give a partial answer to the open problem posed by the authors in 2024.

Keywords

transitive permutation group , movement , action , non-identity element , p-group

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.
• [2] M. Alaeiyan, M. A. Shabeeb and M. Akbarizadeh, Transitive permutation groups with elements of movement m or m − 2, Hacet. J. Math. Stat. 53, 1102-1117, 2024.
• [3] M. Alaeiyan and H. A. Tavallaee, Permutation groups with the same movement, Carpathian J. Math. 25, 147-156, 2009.
• [4] M. Alaeiyan and S. Yoshiara, Permutation groups of minimal movement, Arch. Math. 85, 211-226, 2005.
• [5] J. R. Cho, P. S. Kim and C. E. Praeger, The maximal number of orbits of a permutation groups with bounded movement, J. Algebra, 214, 625-630, 1999.
• [6] A. Hassani, M. Alaeiyan, E. I. Khukhro and C. E. Praeger, Transitive permutation groups with bounded movement having maximal degree, J. Algebra 214, 317-337, 1999.
• [7] H. L. Liu and L. Z. Lu, Transitive permutation groups with elements of movement m or m − 3, (to appear).
• [8] A. Mann and C. E. Praeger, Transitive permutation groups of minimal movement, J. Algebra, 181, 903-911, 1996.
• [9] C. E. Praeger, On permutation groups with bounded movement, J. Algebra, 144, 436-442, 1991.
• [10] C. E. Praeger, Movement and separation of subsets of points under group action, J. Lond. Math. Soc. 56 (2), 519-528, 1997.