On finite totally 2-closed groups

TL;DR

This paper classifies finite soluble totally 2-closed groups, explores properties of their Fitting subgroups, and characterizes minimal insoluble examples with non-trivial Fitting subgroups.

Contribution

It provides a classification of finite soluble totally 2-closed groups and describes the structure of minimal insoluble totally 2-closed groups.

Findings

Fitting subgroup of a totally 2-closed group is also totally 2-closed.

Finite soluble totally 2-closed groups are classified.

Minimal insoluble totally 2-closed groups have a specific shape involving a cyclic center and a nonabelian minimal normal subgroup.

Abstract

An abstract group $G$ is called totally $2$-closed if $H=H^{(2),\Omega}$ for any set $\Omega$ with $G\cong H\leq{\rm Sym}(\Omega)$, where $H^{(2),\Omega}$ is the largest subgroup of ${\rm Sym}(\Omega)$ whose orbits on $\Omega\times\Omega$ are the same orbits of $H$. In this paper, we classify the finite soluble totally $2$-closed groups. We also prove that the Fitting subgroup of a totally $2$-closed group is a totally $2$-closed group. Finally, we prove that a finite insoluble totally $2$-closed group $G$ of minimal order with non-trivial Fitting subgroup has shape $Z\cdot X$, with $Z=Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian.