Handbook of Computational Group Theory

by Derek F. Holt; Bettina Eick; Eamonn A. O'Brien

• ISBN13: 9781420035216
• ISBN10: 1420035215
• Format: Nonspecific Binding
• Copyright: 2005-01-13
• Publisher: Taylor & Francis

Summary

The origins of computation group theory (CGT) date back to the late 19th and early 20th centuries. Since then, the field has flourished, particularly during the past 30 to 40 years, and today it remains a lively and active branch of mathematics. The Handbook of Computational Group Theory offers the first complete treatment of all the fundamental methods and algorithms in CGT presented at a level accessible even to advanced undergraduate students. It develops the theory of algorithms in full detail and highlights the connections between the different aspects of CGT and other areas of computer algebra. While acknowledging the importance of the complexity analysis of CGT algorithms, the authors' primary focus is on algorithms that perform well in practice rather than on those with the best theoretical complexity. Throughout the book, applications of all the key topics and algorithms to areas both within and outside of mathematics demonstrate how CGT fits into the wider world of mathematics and science. The authors include detailed pseudocode for all of the fundamental algorithms, and provide detailed worked examples that bring the theorems and algorithms to life.

Table of Contents

History of Computational Group Theory

Background Materials

Fundamentals

Group Actions

Series

Presentation of Groups

Presentation of Subgroups

Abelian Group Presentations

Representation Theory, Modules, Extension, Derivations, and Complements

Field Theory

Representing Groups on a Computer

Representing Groups on Computers

The Use of Random Methods in CGT

Some Structural Calculators

Computing with Homorphisms

Computation in Finite Permutation Groups

The Calculation of Orbits and Stabilizers

Testing for Alt (W) and Sym (W)

Finding Block Systems

Bases and Strong Generating Sets

Homomorphisms from Permutation Groups

Backtrack Searches

Sylow Subgroups, P-cores, and the Solvable Radical

Applications

Coset Enumeration

The Basic Procedure

Strategies for Coset Enumeration

Presentations of Subgroups

Finding All Subgroups

Finding All Subgroups Up to a Given Index

Applications

Presentation of Given Groups

Finding a Presentation of a Given Group

Finding a Presentation of a Strong Generating Set

The Sims 'Verify' Algorithm

Representations, Cohomology, and Characters

Computation in Finite Fields

Elemetary Computational Linear Algebra

Factorizing Polynomials Over Finite Fields

Testing KG-Models for Irreducibility - The Meataxe

Related Computations

Cohomology

Computing Character Tables

Structural Investigation of Matrix Groups

Computation With Polycyclic Groups

Polycyclic Presentations

Examples of Polycyclic Groups

Subgroups and Membership Testing

Factor Groups and Homomorphisms

Subgroup Series

Orbit-Stabilizer Methods

Complements and Extensions

Intersections, Centralizers, and Normalizers

Automorphism Groups

The Structure of Finite Solvable Groups

Computing Quotients of Finitely Presented Groups

Finite Quotients and Automorphism Groups of Finite Groups

Abelian Quotients

Practical Computation of the HNF and SNF

P-Quotients of Finitely-Presented Groups

Advanced Computations in Finite Groups

Some Useful Subgroups

Computing Composition and Chief Series

Applications of the Solvable Radical Method

Computing the Subgroups of a Finite Group

Appication - Enumerating Finite Unlabelled Structures

Libraries and Databases

Primitive Permutation Groups

Transitive Permutation Groups

Perfect Groups

The Small Groups Library

Crystallorgraphic Groups

Other Databases

Rewriting Systems

Monoid Systems

Rewriting Systems

Rewriting Systems in Monoids and Groups

Rewriting Systems for Polycyclic Groups

Verifying Nilpotency

Applications

Finite State Automata and Automatic Groups

Finite State Automata

Automatic Groups

The Algorithm to Compute Shortlex Automatic Structures

Related Algorithms

Applications

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