Notes on finite totally $2$-closed permutation groups

TL;DR

This paper investigates properties of finite totally 2-closed groups, establishing new results about their structure and providing simplified proofs of existing theorems in the context of permutation groups.

Contribution

It proves that the 2-closure of a normal subgroup in a faithful G-set behaves predictably and shows that abelian normal subgroups are cyclic in totally 2-closed groups, also characterizing nilpotent groups with coprime factors.

Findings

The 2-closure of a normal subgroup in a faithful G-set is preserved.

Abelian normal subgroups of finite totally 2-closed groups are cyclic.

Finite nilpotent groups with coprime totally 2-closed factors are totally 2-closed.

Abstract

Let $N$ be a normal subgroup of a finite group $G$. For a faithful $N$-set $\Delta$, applying the university embedding theorem one can construct a faithful $G$-set $\Omega$. In this short note, it is proved that if the $2$-closure of $N$ in $\Omega$ is equal to $N$, then the $2$-closure of $N$ in $\Delta$ is also equal to $N$; in addition, it is proved that any abelian normal subgroup of a finite totally $2$-closed group is cyclic; finally, it is proved that if a finite nilpotent group is a direct of two nilpotent subgroups where the two factors have coprime orders and both of them are totally 2-closed then G is totally $2$-closed. As corollaries, several well-known results on finite totally 2-closed groups are reproved in more simple ways.