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

Authors : Mehdi Alaeiyan, Murtadha Shabeeb, Masoumeh Akbarizadeh

Pages : 1102-1117

Doi:10.15672/hujms.1223815

View : 70 | Download : 127

Publication Date : 2024-08-27

Article Type : Research Paper

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