Finding direct product decompositions in polynomial time

TL;DR

This paper presents a polynomial-time algorithm for decomposing permutation groups into direct products of indecomposable subgroups, using algebraic methods involving bilinear maps and rings.

Contribution

It introduces a novel polynomial-time algorithm that efficiently finds direct product decompositions of permutation groups using algebraic characterizations.

Findings

Algorithm runs in polynomial time

Applicable to permutation groups and their quotients

Uses algebraic structures like bilinear maps and rings

Abstract

A polynomial-time algorithm is produced which, given generators for a group of permutations on a finite set, returns a direct product decomposition of the group into directly indecomposable subgroups. The process uses bilinear maps and commutative rings to characterize direct products of p-groups of class 2 and reduces general groups to p-groups using group varieties. The methods apply to quotients of permutation groups and operator groups as well.