Handbook of computational group theory

Derek F. Holt, Bettina Eick, Eamonn A. O’Brien1584883723, 9781584883722

——————-Description——————– 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. ———————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 ———————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