[FORTHCOMING] Permutation groups with two orbits having constant movement

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 $\Gamma^{g}-\Gamma|$ is bounded, for $g\in G$, the movement of $g$ is defined as move $(g):=\max{|\Gamma^{g}-\Gamma|}$ over all subsets $\Gamma$ of $\Omega$, and move $(G)$ is defined as the maximum of move $(g)$ over all non-identity elements of $g\in G$. Suppose that $G$ is not a 2-group. It was shown by Praeger that $|\Omega|\leqslant\lceil\frac{2mp}{p-1}\rceil+t-1$, where $t$ is the number of $G$-orbits on $\Omega$ and $p$ is the least odd prime dividing $|G|$. In this paper, we classify all permutation groups with maximum possible degree $|\Omega|=\lceil\frac{2mp}{p-1}\rceil+t-1$ for $t=2$, in which every non-identity element has constant movement $m$.

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 $\Gamma^{g}-\Gamma|$ is bounded, for $g\in G$, the movement of $g$ is defined as move $(g):=\max{|\Gamma^{g}-\Gamma|}$ over all subsets $\Gamma$ of $\Omega$, and move $(G)$ is defined as the maximum of move $(g)$ over all non-identity elements of $g\in G$. Suppose that $G$ is not a 2-group. It was shown by Praeger that $|\Omega|\leqslant\lceil\frac{2mp}{p-1}\rceil+t-1$, where $t$ is the number of $G$-orbits on $\Omega$ and $p$ is the least odd prime dividing $|G|$. In this paper, we classify all permutation groups with maximum possible degree $|\Omega|=\lceil\frac{2mp}{p-1}\rceil+t-1$ for $t=2$, in which every non-identity element has constant movement $m$.

Intransitive permutation groups orbits bounded movement constant movement.

Intransitive permutation groups orbits bounded movement constant movement