Handbook of computational group theory:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2005
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| Series: | Discrete mathematics and its applications
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 514 S. |
| ISBN: | 1584883723 |
Staff View
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| adam_text | Titel: Handbook of computational group theory
Autor: Holt, Derek F.
Jahr: 2005
Contents
Notation and displayed procedures xvi
1 A Historical Review of Computational Group Theory 1
2 Background Material 9
2.1 Fundamentals........................... 9
2.1.1 Definitions......................... 9
2.1.2 Subgroups......................... 11
2.1.3 Cyclic and dihedral groups ............... 12
2.1.4 Generators ........................ 13
2.1.5 Examples ? permutation groups and matrix groups . 13
2.1.6 Normal subgroups and quotient groups......... 14
2.1.7 Homomorphisms and the isomorphism theorems ... 15
2.2 Group actions........................... 17
2.2.1 Definition and examples................. 17
2.2.2 Orbits and stabilizers .................. 19
2.2.3 Conjugacy, normalizers, and centralizers........ 20
2.2.4 Sylow s theorems..................... 21
2.2.5 Transitivity and primitivity............... 22
2.3 Series................................ 26
2.3.1 Simple and characteristically simple groups...... 26
2.3.2 Series ........................... 27
2.3.3 The derived series and solvable groups......... 27
2.3.4 Central series and nilpotent groups........... 29
2.3.5 The socle of a finite group................ 31
2.3.6 The Frattini subgroup of a group............ 32
2.4 Presentations of groups...................... 33
2.4.1 Free groups........................ 33
2.4.2 Group presentations................... 36
2.4.3 Presentations of group extensions............ 38
2.4.4 Tietze transformations.................. 40
2.5 Presentations of subgroups.................... 41
2.5.1 Subgroup presentations on Schreier generators .... 41
2.5.2 Subgroup presentations on a general generating set . . 44
2.6 Abelian group presentations................... 46
2.7 Representation theory, modules, extensions, derivations, and
complements............................ 48
2.7.1 The terminology of representation theory....... 49
2.7.2 Semidirect products, complements, derivations, and
first cohomology groups................. 50
2.7.3 Extensions of modules and the second cohomology
group........................... 52
2.7.4 The actions of automorphisms on cohomology groups . 54
2.8 Field theory............................ 56
2.8.1 Field extensions and splitting fields........... 56
2.8.2 Finite fields........................ 58
2.8.3 Conway polynomials................... 59
Representing Groups on a Computer 61
3.1 Representing groups on computers................ 61
3.1.1 The fundamental representation types......... 61
3.1.2 Computational situations................ 62
3.1.3 Straight-line programs.................. 64
3.1.4 Black-box groups..................... 65
3.2 The use of random methods in CGT............... 67
3.2.1 Randomized algorithms................. 67
3.2.2 Finding random elements of groups........... 69
3.3 Some structural calculations................... 72
3.3.1 Powers and orders of elements.............. 72
3.3.2 Normal closure...................... 73
3.3.3 The commutator subgroup, derived series, and lower
central series....................... 73
3.4 Computing with homomorphisms................ 74
3.4.1 Defining and verifying group homomorphisms..... 74
3.4.2 Desirable facilities.................... 75
Computation in Finite Permutation Groups 77
4.1 The calculation of orbits and stabilizers............. 77
4.1.1 Schreier vectors...................... 79
4.2 Testing for Alt(ft) and Sym(ft).................. 81
4.3 Finding block systems....................... 82
4.3.1 Introduction........................ 82
4.3.2 The Atkinson algorithm................. 83
4.3.3 Implementation of the class merging process...... 85
4.4 Bases and strong generating sets................. 87
4.4.1 Definitions......................... 87
4.4.2 The Schreier-Sims algorithm............... 90
4.4.3 Complexity and implementation issues......... 93
4.4.4 Modifying the strong generating set ? shallow Schreier
trees............................ 95
XI
4.4.5 The random Schreier-Sims method........... 97
4.4.6 The solvable BSGS algorithm.............. 98
4.4.7 Change of base......................102
4.5 Homomorphisms from permutation groups...........105
4.5.1 The induced action on a union of orbits........ 105
4.5.2 The induced action on a block system......... 106
4.5.3 Homomorphisms between permutation groups..... 107
4.6 Backtrack searches ........................ 108
4.6.1 Searching through the elements of a group....... 110
4.6.2 Pruning the tree..................... 113
4.6.3 Searching for subgroups and coset representatives . . . 114
4.6.4 Automorphism groups of combinatorial structures and
partitions......................... 118
4.6.5 Normalizers and centralizers............... 121
4.6.6 Intersections of subgroups................ 124
4.6.7 Transversals and actions on cosets........... 126
4.6.8 Finding double coset representatives.......... 131
4.7 Sylow subgroups, p-eores, and the solvable radical....... 132
4.7.1 Reductions involving intransitivity and imprimitivity . 133
4.7.2 Computing Sylow subgroups............... 134
4.7.3 A result on quotient groups of permutation groups . . 137
4.7.4 Computing the p-core.................. 138
4.7.5 Computing the solvable radical............. 140
4.7.6 Nonabelian regular normal subgroups ......... 141
4.8 Applications............................ 143
4.8.1 Card shuffling.......................144
4.8.2 Graphs, block designs, and error-correcting codes . . . 145
4.8.3 Diameters of Cayley graphs...............147
4.8.4 Processor interconnection networks...........148
Coset Enumeration 149
5.1 The basic procedure........................ 150
5.1.1 Coset tables and their properties............ 151
5.1.2 Denning and scanning.................. 152
5.1.3 Coincidences ....................... 156
5.2 Strategies for coset enumeration................. 162
5.2.1 The relator-based method................ 162
5.2.2 The coset table-based method.............. 165
5.2.3 Compression and standardization............ 167
5.2.4 Recent developments and examples........... 168
5.2.5 Implementation issues.................. 170
5.2.6 The use of coset enumeration in practice........ 171
5.3 Presentations of subgroups.................... 173
5.3.1 Computing a presentation on Schreier generators . . . 173
5.3.2 Computing a presentation on the user generators . . . 178
xu
5.3.3 Simplifying presentations ................184
5.4 Finding all subgroups up to a given index............188
5.4.1 Coset tables for a group presentation..........189
5.4.2 Details of the procedure.................190
5.4.3 Variations and improvements..............196
5.5 Applications............................198
Presentations of Given Groups 199
6.1 Finding a presentation of a given group.............199
6.2 Finding a presentation on a set of strong generators......205
6.2.1 The known BSGS case..................205
6.2.2 The Todd-Coxeter-Schreier-Sims algorithm......207
6.3 The Sims Verify algorithm ...................208
6.3.1 The single-generator case................209
6.3.2 The general case.....................213
6.3.3 Examples.........................217
Representation Theory, Cohomology, and Characters 219
7.1 Computation in finite fields.................... 220
7.2 Elementary computational linear algebra............ 221
7.3 Factorizing polynomials over finite fields ............ 226
7.3.1 Reduction to the squarefree case............ 228
7.3.2 Reduction to constant-degree irreducibles....... 229
7.3.3 The constant-degree case ................ 229
7.4 Testing iiTG-modules for irreducibility ? the Meataxe..... 230
7.4.1 The Meataxe algorithm................. 230
7.4.2 Proof of correctness ................... 234
7.4.3 The Ivanyos-Lux extension ............... 235
7.4.4 Actions on submodules and quotient modules..... 235
7.4.5 Applications........................ 236
7.5 Related computations....................... 237
7.5.1 Testing modules for absolute irreducibility....... 237
7.5.2 Finding module homomorphisms............ 241
7.5.3 Testing irreducible modules for isomorphism...... 244
7.5.4 Application ? invariant bilinear forms......... 245
7.5.5 Finding all irreducible representations over a finite
field............................ 246
7.6 Cohomology............................ 248
7.6.1 Computing first cohomology groups .......... 249
7.6.2 Deciding whether an extension splits.......... 253
7.6.3 Computing second cohomology groups......... 254
7.7 Computing character tables ................... 255
7.7.1 The basic method .................... 256
7.7.2 Working modulo a prime................. 257
7.7.3 Further improvements.................. 260
xiu
7.8 Structural investigation of matrix groups............264
7.8.1 Methods based on bases and strong generating sets . . 264
7.8.2 Computing in large-degree matrix groups.......268
8 Computation with Polycyclic Groups 273
8.1 Polycyclic presentations......................274
8.1.1 Polycyclic sequences...................274
8.1.2 Polycyclic presentations and consistency........278
8.1.3 The collection algorithm.................280
8.1.4 Changing the presentation................284
8.2 Examples of polycyclic groups..................286
8.2.1 Abelian, nilpotent, and supersolvable groups .....286
8.2.2 Infinite polycyclic groups and number fields......288
8.2.3 Application ? crystallographic groups.........289
8.3 Subgroups and membership testing...............290
8.3.1 Induced polycyclic sequences..............291
8.3.2 Canonical polycyclic sequences.............296
8.4 Factor groups and homomorphisms...............298
8.4.1 Factor groups.......................298
8.4.2 Homomorphisms.....................299
8.5 Subgroup series..........................300
8.6 Orbit-stabilizer methods.....................302
8.7 Complements and extensions...................304
8.7.1 Complements and the first cohomology group.....304
8.7.2 Extensions and the second cohomology group.....307
8.8 Intersections, centralizers, and normalizers...........311
8.8.1 Intersections........................311
8.8.2 Centralizers........................313
8.8.3 Normalizers........................314
8.8.4 Conjugacy problems and conjugacy classes.......316
8.9 Automorphism groups.......................317
8.10 The structure of finite solvable groups..............320
8.10.1 Sylow and Hall subgroups................320
8.10.2 Maximal subgroups....................322
9 Computing Quotients of Finitely Presented Groups 325
9.1 Finite quotients and automorphism groups of finite groups . . 326
9.1.1 Description of the algorithm...............326
9.1.2 Performance issues....................332
9.1.3 Automorphism groups of finite groups.........333
9.2 Abelian quotients.........................335
9.2.1 The linear algebra of a free abelian group.......335
9.2.2 Elementary row operations ...............336
9.2.3 The Hermite normal form................337
XIV
9.2.4 Elementary column matrices and the Smith normal
form............................341
9.3 Practical computation of the HNF and SNF..........347
9.3.1 Modular techniques....................347
9.3.2 The use of norms and row reduction techniques .... 349
9.3.3 Applications........................352
9.4 p-quotients of finitely presented groups.............353
9.4.1 Power-conjugate presentations.............. 353
9.4.2 The p-quotient algorithm ................ 355
9.4.3 Other quotient algorithms................ 364
9.4.4 Generating descriptions of p-groups........... 364
9.4.5 Testing finite p-groups for isomorphism ........ 371
9.4.6 Automorphism groups of finite p-groups........ 371
9.4.7 Applications........................ 372
10 Advanced Computations in Finite Groups 375
10.1 Some useful subgroups...................... 376
10.1.1 Definition of the subgroups............... 376
10.1.2 Computing the subgroups ? initial reductions .... 377
10.1.3 The O Nan-Scott theorem................ 378
10.1.4 Finding the socle factors - the primitive case..... 379
10.2 Computing composition and chief series............. 381
10.2.1 Refining abelian sections................. 381
10.2.2 Identifying the composition factors........... 382
10.3 Applications of the solvable radical method........... 383
10.4 Computing the subgroups of a finite group........... 385
10.4.1 Identifying the TF-factor ................ 386
10.4.2 Lifting subgroups to the next layer........... 387
10.5 Application - enumerating finite unlabelled structures..... 390
11 Libraries and Databases 393
11.1 Primitive permutation groups..................394
11.1.1 Affine primitive permutation groups..........395
11.1.2 Nonaffine primitive permutation groups........396
11.2 Transitive permutation groups..................397
11.2.1 Summary of the method.................397
11.2.2 Applications........................399
11.3 Perfect groups...........................400
11.4 The small groups library.....................402
11.4.1 The Frattini extension method.............404
11.4.2 A random isomorphism test...............405
11.5 Crystallographic groups......................407
11.6 The ATLAS of Finite Groups .................409
12 Rewriting Systems and the Knuth-Bendix Completion
Process 411
12.1 Monoid presentations.......................412
12.1.1 Monoids and semigroups.................412
12.1.2 Free monoids and monoid presentations........415
12.2 Rewriting systems.........................417
12.3 Rewriting systems in monoids and groups............423
12.4 Rewriting systems for polycyclic groups.............426
12.5 Verifying nilpotency........................429
12.6 Applications............................431
13 Finite State Automata and Automatic Groups 433
13.1 Finite state automata.......................434
13.1.1 Definitions and examples ................434
13.1.2 Enumerating and counting the language of a dfa ... 437
13.1.3 The use of fsa in rewriting systems...........439
13.1.4 Word-acceptors......................441
13.1.5 2-variable fsa.......................442
13.1.6 Operations on finite state automata..........442
13.1.6.1 Making an fsa deterministic...........443
13.1.6.2 Minimizing an fsa................444
13.1.6.3 Testing for language equality..........446
13.1.6.4 Negation, union, and intersection........447
13.1.6.5 Concatenation and star.............447
13.1.7 Existential quantification ................448
13.2 Automatic groups.........................451
13.2.1 Definitions, examples, and background.........451
13.2.2 Word-differences and word-difference automata .... 453
13.3 The algorithm to compute the shortlex automatic structures . 456
13.3.1 Step 1...........................457
13.3.2 Step 2 and word reduction................459
13.3.3 Step 3...........................460
13.3.4 Step 4...........................462
13.3.5 Step 5...........................464
13.3.6 Comments on the implementation and examples . . . 466
13.4 Related algorithms ........................468
13.5 Applications............................469
References 471
Index of Displayed Procedures .....................497
Author Index...............................499
Subject Index...............................503
|
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| record_format | marc |
| series2 | Discrete mathematics and its applications |
| spelling | Holt, Derek F. Verfasser aut Handbook of computational group theory Derek F. Holt ; Bettina Eick ; Eamonn A. O'Brien Boca Raton [u.a.] Chapman & Hall/CRC 2005 XVI, 514 S. txt rdacontent n rdamedia nc rdacarrier Discrete mathematics and its applications Dataprocessing gtt Groepentheorie gtt Groupes finis - Informatique Groupes, Théorie combinatoire des - Informatique Groupes, Théorie des - Informatique aGroup theory xData processing aFinite groups xData processing aCombinatorial group theory xData processing Algorithmische Gruppentheorie (DE-588)4705829-8 gnd rswk-swf Algorithmische Gruppentheorie (DE-588)4705829-8 s DE-604 Eick, Bettina Verfasser aut O'Brien Eamonn A. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013071804&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Holt, Derek F. Eick, Bettina O'Brien Eamonn A. Handbook of computational group theory Dataprocessing gtt Groepentheorie gtt Groupes finis - Informatique Groupes, Théorie combinatoire des - Informatique Groupes, Théorie des - Informatique aGroup theory xData processing aFinite groups xData processing aCombinatorial group theory xData processing Algorithmische Gruppentheorie (DE-588)4705829-8 gnd |
| subject_GND | (DE-588)4705829-8 |
| title | Handbook of computational group theory |
| title_auth | Handbook of computational group theory |
| title_exact_search | Handbook of computational group theory |
| title_full | Handbook of computational group theory Derek F. Holt ; Bettina Eick ; Eamonn A. O'Brien |
| title_fullStr | Handbook of computational group theory Derek F. Holt ; Bettina Eick ; Eamonn A. O'Brien |
| title_full_unstemmed | Handbook of computational group theory Derek F. Holt ; Bettina Eick ; Eamonn A. O'Brien |
| title_short | Handbook of computational group theory |
| title_sort | handbook of computational group theory |
| topic | Dataprocessing gtt Groepentheorie gtt Groupes finis - Informatique Groupes, Théorie combinatoire des - Informatique Groupes, Théorie des - Informatique aGroup theory xData processing aFinite groups xData processing aCombinatorial group theory xData processing Algorithmische Gruppentheorie (DE-588)4705829-8 gnd |
| topic_facet | Dataprocessing Groepentheorie Groupes finis - Informatique Groupes, Théorie combinatoire des - Informatique Groupes, Théorie des - Informatique aGroup theory xData processing aFinite groups xData processing aCombinatorial group theory xData processing Algorithmische Gruppentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013071804&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT holtderekf handbookofcomputationalgrouptheory AT eickbettina handbookofcomputationalgrouptheory AT obrieneamonna handbookofcomputationalgrouptheory |