Handbook of Computational Gr Theory 1st Edition by Derek Holt, Bettina Eick, Eamonn Brien 1040069172 9781040069172

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ISBN 10: 1040069172 

ISBN 13: 9781040069172

Author: Derek F. Holt, Bettina Eick, Eamonn A. O’Brien 

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 fundame

Handbook of Computational Gr Theory 1st Table of contents: 

1 A Historical Review of Computational Group Theory
2 Background Material
2.1 Fundamentals
2.1.1 Definitions
2.1.2 Subgroups
2.1.3 Cyclic and dihedral groups
2.1.4 Generators
2.1.5 Examples — permutation groups and matrix groups
2.1.6 Normal subgroups and quotient groups
2.1.7 Homomorphisms and the isomorphism theorems
2.2 Group actions
2.2.1 Definition and examples
2.2.2 Orbits and stabilizers
2.2.3 Conjugacy, normalizers, and centralizers
2.2.4 Sylow’s theorems
2.2.5 Transitivity and primitivity
2.3 Series
2.3.1 Simple and characteristically simple groups
2.3.2 Series
2.3.3 The derived series and solvable groups
2.3.4 Central series and nilpotent groups
2.3.5 The socle of a finite group
2.3.6 The Frattini subgroup of a group
2.4 Presentations of groups
2.4.1 Free groups
2.4.2 Group presentations
2.4.3 Presentations of group extensions
2.4.4 Tietze transformations
2.5 Presentations of subgroups
2.5.1 Subgroup presentations on Schreier generators
2.5.2 Subgroup presentations on a general generating set
2.6 Abelian group presentations
2.7 Representation theory, modules, extensions, derivations, and complements
2.7.1 The terminology of representation theory
2.7.2 Semidirect products, complements, derivations, and first cohomology groups
2.7.3 Extensions of modules and the second cohomology group
2.7.4 The actions of automorphisms on cohomology groups
2.8 Field theory
2.8.1 Field extensions and splitting fields
2.8.2 Finite fields
2.8.3 Conway polynomials
3 Representing Groups on a Computer
3.1 Representing groups on computers
3.1.1 The fundamental representation types
3.1.2 Computational situations
3.1.3 Straight-line programs
3.1.4 Black-box groups
3.2 The use of random methods in CGT
3.2.1 Randomized algorithms
3.2.2 Finding random elements of groups
3.3 Some structural calculations
3.3.1 Powers and orders of elements
3.3.2 Normal closure
3.3.3 The commutator subgroup, derived series, and lower central series
3.4 Computing with homomorphisms
3.4.1 Defining and verifying group homomorphisms
3.4.2 Desirable facilities
4 Computation in Finite Permutation Groups
4.1 The calculation of orbits and stabilizers
4.1.1 Schreier vectors
4.2 Testing for Alt(Ω) and Sym(Ω)
4.3 Finding block systems
4.3.1 Introduction
4.3.2 The Atkinson algorithm
4.3.3 Implementation of the class merging process
4.4 Bases and strong generating sets
4.4.1 Definitions
4.4.2 The Schreier-Sims algorithm
4.4.3 Complexity and implementation issues
4.4.4 Modifying the strong generating set — shallow Schreier trees
4.4.5 The random Schreier-Sims method
4.4.6 The solvable BSGS algorithm
4.4.7 Change of base
4.5 Homomorphisms from permutation groups
4.5.1 The induced action on a union of orbits
4.5.2 The induced action on a block system
4.5.3 Homomorphisms between permutation groups
4.6 Backtrack searches
4.6.1 Searching through the elements of a group
4.6.2 Pruning the tree
4.6.3 Searching for subgroups and coset representatives
4.6.4 Automorphism groups of combinatorial structures and partitions
4.6.5 Normalizers and centralizers
4.6.6 Intersections of subgroups
4.6.7 Transversals and actions on cosets
4.6.8 Finding double coset representatives
4.7 Sylow subgroups, p-cores, and the solvable radical
4.7.1 Reductions involving intransitivity and imprimitivity
4.7.2 Computing Sylow subgroups
4.7.3 A result on quotient groups of permutation groups
4.7.4 Computing the p-core
4.7.5 Computing the solvable radical
4.7.6 Nonabelian regular normal subgroups
4.8 Applications
4.8.1 Card shuffling
4.8.2 Graphs, block designs, and error-correcting codes
4.8.3 Diameters of Cayley graphs
4.8.4 Processor interconnection networks
5 Coset Enumeration
5.1 The basic procedure
5.1.1 Coset tables and their properties
5.1.2 Defining and scanning
5.1.3 Coincidences
5.2 Strategies for coset enumeration
5.2.1 The relator-based method
5.2.2 The coset table-based method
5.2.3 Compression and standardization
5.2.4 Recent developments and examples
5.2.5 Implementation issues
5.2.6 The use of coset enumeration in practice
5.3 Presentations of subgroups
5.3.1 Computing a presentation on Schreier generators
5.3.2 Computing a presentation on the user generators
5.3.3 Simplifying presentations
5.4 Finding all subgroups up to a given index
5.4.1 Coset tables for a group presentation
5.4.2 Details of the procedure
5.4.3 Variations and improvements
5.5 Applications
6 Presentations of Given Groups
6.1 Finding a presentation of a given group
6.2 Finding a presentation on a set of strong generators
6.2.1 The known BSGS case
6.2.2 The Todd-Coxeter-Schreier-Sims algorithm
6.3 The Sims ‘Verify’ algorithm
6.3.1 The single-generator case
6.3.2 The general case
6.3.3 Examples
7 Representation Theory, Cohomology, and Characters
7.1 Computation in finite fields
7.2 Elementary computational linear algebra
7.3 Factorizing polynomials over finite fields
7.3.1 Reduction to the squarefree case
7.3.2 Reduction to constant-degree irreducibles
7.3.3 The constant-degree case
7.4 Testing KG-modules for irreducibility — the Meataxe
7.4.1 The Meataxe algorithm
7.4.2 Proof of correctness
7.4.3 The Ivanyos-Lux extension
7.4.4 Actions on submodules and quotient modules
7.4.5 Applications
7.5 Related computations
7.5.1 Testing modules for absolute irreducibility
7.5.2 Finding module homomorphisms
7.5.3 Testing irreducible modules for isomorphism
7.5.4 Application — invariant bilinear forms
7.5.5 Finding all irreducible representations over a finite field
7.6 Cohomology
7.6.1 Computing first cohomology groups
7.6.2 Deciding whether an extension splits
7.6.3 Computing second cohomology groups
7.7 Computing character tables
7.7.1 The basic method
7.7.2 Working modulo a prime
7.7.3 Further improvements
7.8 Structural investigation of matrix groups
7.8.1 Methods based on bases and strong generating sets
7.8.2 Computing in large-degree matrix groups
8 Computation with Polycyclic Groups
8.1 Polycyclic presentations
8.1.1 Polycyclic sequences
8.1.2 Polycyclic presentations and consistency
8.1.3 The collection algorithm
8.1.4 Changing the presentation
8.2 Examples of polycyclic groups
8.2.1 Abelian, nilpotent, and supersolvable groups
8.2.2 Infinite polycyclic groups and number fields
8.2.3 Application — crystallographic groups
8.3 Subgroups and membership testing
8.3.1 Induced polycyclic sequences
8.3.2 Canonical polycyclic sequences
8.4 Factor groups and homomorphisms
8.4.1 Factor groups
8.4.2 Homomorphisms
8.5 Subgroup series
8.6 Orbit-stabilizer methods
8.7 Complements and extensions
8.7.1 Complements and the first cohomology group
8.7.2 Extensions and the second cohomology group
8.8 Intersections, centralizers, and normalizers
8.8.1 Intersections
8.8.2 Centralizers
8.8.3 Normalizers
8.8.4 Conjugacy problems and conjugacy classes
8.9 Automorphism groups
8.10 The structure of finite solvable groups
8.10.1 Sylow and Hall subgroups
8.10.2 Maximal subgroups
9 Computing Quotients of Finitely Presented Groups
9.1 Finite quotients and automorphism groups of finite groups
9.1.1 Description of the algorithm
9.1.2 Performance issues
9.1.3 Automorphism groups of finite groups
9.2 Abelian quotients
9.2.1 The linear algebra of a free abelian group
9.2.2 Elementary row operations
9.2.3 The Hermite normal form
9.2.4 Elementary column matrices and the Smith normal form
9.3 Practical computation of the HNF and SNF
9.3.1 Modular techniques
9.3.2 The use of norms and row reduction techniques
9.3.3 Applications
9.4 p-quotients of finitely presented groups
9.4.1 Power-conjugate presentations
9.4.2 The p-quotient algorithm
9.4.3 Other quotient algorithms
9.4.4 Generating descriptions of p-groups
9.4.5 Testing finite p-groups for isomorphism
9.4.6 Automorphism groups of finite p-groups
9.4.7 Applications
10 Advanced Computations in Finite Groups
10.1 Some useful subgroups
10.1.1 Definition of the subgroups
10.1.2 Computing the subgroups — initial reductions
10.1.3 The O’Nan-Scott theorem
10.1.4 Finding the socle factors — the primitive case
10.2 Computing composition and chief series
10.2.1 Refining abelian sections
10.2.2 Identifying the composition factors
10.3 Applications of the solvable radical method
10.4 Computing the subgroups of a finite group
10.4.1 Identifying the TF-factor
10.4.2 Lifting subgroups to the next layer
10.5 Application – enumerating finite unlabelled structures
11 Libraries and Databases
11.1 Primitive permutation groups
11.1.1 Affine primitive permutation groups
11.1.2 Nonaffine primitive permutation groups
11.2 Transitive permutation groups
11.2.1 Summary of the method
11.2.2 Applications
11.3 Perfect groups
11.4 The small groups library
11.4.1 The Frattini extension method
11.4.2 A random isomorphism test
11.5 Crystallographic groups
11.6 The “ATLAS of Finite Groups”
12 Rewriting Systems and the Knuth-Bendix Completion Process
12.1 Monoid presentations
12.1.1 Monoids and semigroups
12.1.2 Free monoids and monoid presentations
12.2 Rewriting systems
12.3 Rewriting systems in monoids and groups
12.4 Rewriting systems for polycyclic groups
12.5 Verifying nilpotency
12.6 Applications
13 Finite State Automata and Automatic Groups
13.1 Finite state automata
13.1.1 Definitions and examples
13.1.2 Enumerating and counting the language of a dfa
13.1.3 The use of fsa in rewriting systems
13.1.4 Word-acceptors
13.1.5 2 -variable fsa
13.1.6 Operations on finite state automata
13.1.6.1 Making an fsa deterministic
13.1.6.2 Minimizing an fsa
13.1.6.3 Testing for language equality
13.1.6.4 Negation, union, and intersection
13.1.6.5 Concatenation and star
13.1.7 Existential quantification
13.2 Automatic groups
13.2.1 Definitions, examples, and background
13.2.2 Word-differences and word-difference automata
13.3 The algorithm to compute the shortlex automatic structures
13.3.1 Step 1
13.3.2 Step 2 and word reduction
13.3.3 Step 3
13.3.4 Step 4
13.3.5 Step 5
13.3.6 Comments on the implementation and examples
13.4 Related algorithms
13.5 Applications

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