Subgroups of minimal index in polynomial time

TL;DR

This paper presents a polynomial-time algorithm to compute minimal index subgroups in finite groups, explicitly find such subgroups in permutation groups, and test group actions on trees, advancing computational group theory.

Contribution

It introduces a polynomial algorithm for minimal index subgroup computation and explicit subgroup finding, extending previous theoretical results.

Findings

Polynomial algorithm for minimal index subgroup computation

Explicit subgroup identification in permutation groups

Algorithm for testing group actions on trees

Abstract

Let $G$ be a finite group and let $H$ be a proper subgroup of $G$ of minimal index. By applying an old result of Y. Berkovich, we provide a polynomial algorithm for computing $|G : H|$ for a permutation group $G$. Moreover, we find $H$ explicitly if $G$ is given by a Cayley table. As a corollary, we get an algorithm for testing whether a finite permutation group acts on a tree or not.