Handbook of Computational Group Theory

ISBN-13: 9781584883722 / Angielski / Twarda / 2005 / 536 str.

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

Opis

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 with 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.

Spis treści

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

O autorze

Derek F. Holt, Bettina Eick, Eamonn A. O'Brien