Closures of permutation groups with restricted nonabelian composition factors

TL;DR

This paper proves that the $k$-closure of an $ ext{Alt}(d)$-free permutation group remains $ ext{Alt}(d)$-free for sufficiently large $k$ and $d$, advancing understanding of group closures with restricted composition factors.

Contribution

It establishes that the property of being $ ext{Alt}(d)$-free is preserved under $k$-closure for $k \, ext{at least}\, 4$ and $d \, ext{at least}\, 25$, a new result in computational group theory.

Findings

The $k$-closure of an $ ext{Alt}(d)$-free group is $ ext{Alt}(d)$-free for $k \, ext{≥}\, 4$ and $d \, ext{≥}\, 25.

This result constrains the structure of permutation groups with restricted nonabelian composition factors.

Supports further research in computational group theory and symmetry analysis.

Abstract

Given a permutation group $G$ on a finite set $\Omega$, let $G^{(k)}$ denote the $k$-closure of $G$, that is, the largest permutation group on $\Omega$ having the same orbits in the induced action on $\Omega^k$ as $G$. Recall that a group is $\mathrm{Alt}(d)$-free if it does not contain a section isomorphic to the alternating group of degree $d$. Motivated by some problems in computational group theory, we prove that the $k$-closure of an $\mathrm{Alt}(d)$-free group is again $\mathrm{Alt}(d)$-free for $k \geq 4$ and $d \geq 25$.