The $3$-closure of a solvable permutation group is solvable

TL;DR

This paper proves that for any solvable permutation group, its $m$-closure remains solvable when $m$ is at least 3, extending understanding of the structure of permutation groups.

Contribution

The paper establishes that the $m$-closure of a solvable permutation group is always solvable for all $m \,\geq\, 3$, filling a gap in group theory knowledge.

Findings

The $1$-closure and $2$-closure of a solvable group may not be solvable.

The $m$-closure for $m\geq3$ preserves solvability.

Provides a new insight into the structure of permutation groups.

Abstract

Let $m$ be a positive integer and let $\Omega$ be a finite set. The $m$-closure of $G\leq\operatorname{Sym}(\Omega)$ is the largest permutation group on $\Omega$ having the same orbits as $G$ in its induced action on the Cartesian product $\Omega^m$. The $1$-closure and $2$-closure of a solvable permutation group need not be solvable. We prove that the $m$-closure of a solvable permutation group is always solvable for $m\geq3$.