Transitive permutation groups with elements of movement $m$ or $m-2$

Year 2024, , 1102 - 1117, 27.08.2024

Mehdi Alaeiyan , Murtadha Shabeeb Masoumeh Akbarizadeh

https://doi.org/10.15672/hujms.1223815

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 $m-2$.

Keywords

permutation group, transitive, movement, fixed point free element

References

• [1] M. Alaeiyan and B. Askari, Transitive permutation groups with elements of movement m or m-1, Math. Reports 14 (64), 4 , 317-324, 2012.
• [2] M. Alaeiyan and M. Rezaei, Intransitive permutation groups with bounded movement having maximum degree, Math. Rep. 13 (63), 109-115, 2011.
• [3] M. Alaeiyan and H. Tavallaee, Permutation groups with the same movement, Carpathian J. Math., 147-156, 2009.
• [4] T. Dokchitser, Transitive groups of degree up to 31, accessed on 27 july 2022. URL: Group- Names.org.https://people.maths.bris.ac.uk/ matyd/GroupNames/T31.htmltml.
• [5] B. Fein, W. M. Kantor, and M. Schacher, Relative brauer groups ii, J. reine angew. Math 328, 39-57, 1981.
• [6] G. Group, Gap-groups, algorithms, and programming, version 4.11. 1, 2021. URL: https://www. gap-system. org.
• [7] A. Hassani, M. Alaeiyan(Khayaty), E. Khukhro, and C. E. Praeger, Transitive permutation groups with bounded movement having maximal degree, J. Algebra 214 (1), 317-337, 1999.
• [8] A. Mann and C. E. Praeger, Transitive permutation groups of minimal movement, J. Algebra 181(3), 903-911, 1996.
• [9] C. E. Praeger, On permutation groups with bounded movement, J. Algebra 144 (2), 436-442, 1991.
• [10] J. J. Rotman, An introduction to the theory of groups, Number 3rd ed. Allyn and Bacon, Boston, 1984.
• [11] T. Tsuzuku, Finite groups and finite geometries, volume 78, Cambridge University Press, 1982.