Two-closure of supersolvable permutation group in polynomial time

TL;DR

This paper proves that the 2-closure of a supersolvable permutation group can be computed efficiently in polynomial time, and characterizes its composition factors as cyclic or alternating of prime degree.

Contribution

It introduces a polynomial-time algorithm for computing the 2-closure of supersolvable groups and analyzes the structure of their composition factors.

Findings

2-closure of supersolvable groups is computable in polynomial time

Composition factors of the 2-closure are cyclic or alternating of prime degree

Provides structural insights into the 2-closure of supersolvable groups

Abstract

The $2$-closure $\overline{G}$ of a permutation group $G$ on $\Omega$ is defined to be the largest permutation group on $\Omega$, having the same orbits on $\Omega\times\Omega$ as $G$. It is proved that if $G$ is supersolvable, then $\overline{G}$ can be found in polynomial time in $|\Omega|$. As a byproduct of our technique, it is shown that the composition factors of $\overline{G}$ are cyclic or alternating of prime degree.