Applications of computational tools for finitely presented groups

TL;DR

This paper reviews computational tools and methods for analyzing finitely presented groups, demonstrating how they can determine group properties like solubility and structure through quotient and subgroup analysis.

Contribution

It introduces a general computational approach for finitely presented groups using quotients and subgroups, with practical examples and algorithms.

Findings

Tools can identify group solubility and derived series.

Computational methods provide detailed group structural information.

Algorithms enable analysis of complex finitely presented groups.

Abstract

Computer based techniques for recognizing finitely presented groups are quite powerful. Tools available for this purpose are outlined. They are available both in stand-alone programs and in more comprehensive systems. A general computational approach for investigating finitely presented groups by way of quotients and subgroups is described and examples are presented. The techniques can provide detailed information about group structure. Under suitable circumstances a finitely presented group can be shown to be soluble and its complete derived series can be determined, using what is in effect a soluble quotient algorithm.