Group action-stabilizer graph of group actions of a group on a set

Abstract

In this paper we introduce the group action-stabilizer graph $GAS(G)$ of a group $G$ on a set $X$ with vertex set as the collection of all the group actions of $G$ on $X$, and any two vertices $\varphi$ and $\psi$ are adjacent if and only if the non-trivial subgroups $\cap G_x^{\varphi }$ and $\cap G_x^{\psi }$ of $G$ intersect non-trivially, where $G_x^{\varphi }$ and $G_x^{\psi }$ are two stabilizers of $x$ with respect to the actions $\varphi$ and $\psi$, respectively. We characterize a special subgraph $gas(G)$ of $GAS(G)$ in which the vertex set contains the actions $\varphi$ of $G$ on $X$ such that $\cap G_x^{\varphi }$'s are distinct. We determine the number of group actions within some specific groups and find certain conditions under which $GAS(G)$ is equal to $gas(G)$. We also examine the conditions under which $gas(G)$ and its complement are derived graph for a finite nilpotent group $G$.