Handbook of Computational Group Theory

Series:
Discrete Mathematics and Its Applications
Published:
January 13, 2005 by Chapman and Hall/CRC - 536 Pages
Author(s):
Derek F. Holt, University of Warwick, Coventry, UK; Bettina Eick, Technische Universit¿t, Braunschweig, Germany; Eamonn A. O'Brien, University of Auckland, New Zealand

Purchasing Options

Hardback
$119.95
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ISBN 9781584883722
Cat# C3723
 

Features

  • Provides the first reasonably accessible, comprehensive presentation of computational group theory
  • Summarizes state-of-the-art methods and results, including pointers to the literature for the principal areas of CGT
  • Incorporates the full underlying theory and correctness proofs of basic algorithms, and presents those algorithms in pseudocode
  • Includes a chapter on the precomputed stored libraries and databases of groups and character tables now publicly available

    • 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

    Editorial Reviews

    "This is a book I am very happy to have, both for the choice of content and the quality of exposition. Its subject is a very complete and up-to-date review of computational group theory. …All together, the book contains of a huge amount of information. …I think every mathematician will want this book on his shelf."
    -Mathematics of Computation

    "It will be an indispensable source for any user in this field."

    – G. Kowol, in Monatshefte fur Math, 2007, Vol. 151, No. 3

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