On finite totally 2-closed groups

Abstract

An abstract group $G$ is called totally 2-closed if $H = H^{(2),\Omega}$ for any set $\Omega$ with $G\cong H\leq\textrm{Sym}_\Omega$, where $H^{(2),\Omega}$ is the largest subgroup of symmetric group of $\Omega$ whose orbits on $\Omega\times\Omega$ are the same orbits of $H$. In this paper, we prove that the Fitting subgroup of a totally 2-closed group is a totally 2-closed group. We also conjecture that a finite group $G$ is totally 2-closed if and only if it is cyclic or a direct product of a cyclic group of odd order with a generalized quaternion group. We prove the conjecture in the soluble case, and reduce the general case to groups $G$ of shape $Z\cdot X$, with $Z = Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian