Subgroup Chains and Lagrange Coordinatizations of Finite Permutation Groups

TL;DR

This paper presents a constructive method for hierarchical coordinatizations of finite permutation groups, exploring how subgroup chains create different coordinate systems, supported by computational examples.

Contribution

It introduces a general constructive proof for Lagrange Decompositions of permutation groups, expanding understanding of subgroup chain-based coordinate systems.

Findings

Hierarchical coordinatizations can be systematically constructed.

Subgroup chains yield diverse coordinate systems.

Computational examples validate the theoretical framework.

Abstract

We give a general constructive proof for hierarchical coordinatizations (Lagrange Decompositions) of permutation groups. The generalization originates from the investigation of how the subgroup chains of finite permutation groups yield different coordinate systems. The study is motivated by the practical needs and the verification of an existing computational implementation. Large scale machine calculated examples are also presented.