This handbook covers the whole subject of computational group theory (CGT) at a level suitable for beginning graduate students who have some knowledge of group theory and computer algorithms. It develops the theory of algorithms in full detail, includes complexity analyses whenever possible, and highlights the connections between the different aspects of CGT and with other areas of computer algebra. Several specialist sections provide pointers to the current state of the art in these areas, and all sections include exercises of varying difficulty. For each major collection of algorithms, the book includes a section describing applications both within and outside of group theory.

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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