Leech Computational Problems in Abstract Algebra

Proceedings of a Conference Held at Oxford Under the Auspices of the Science Research Council Atlas Computer Laboratory, 29th August to 2nd September 1967
1. Auflage 2014
ISBN: 978-1-4831-5942-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 412 Seiten, Web PDF

Computational Problems in Abstract Algebra provides information pertinent to the application of computers to abstract algebra. This book discusses combinatorial problems dealing with things like generation of permutations, projective planes, orthogonal latin squares, graphs, difference sets, block designs, and Hadamard matrices. Comprised of 35 chapters, this book begins with an overview of the methods utilized in and results obtained by programs for the investigation of groups. This text then examines the method for establishing the order of a finite group defined by a set of relations satisfied by its generators. Other chapters describe the modification of the Todd-Coxeter coset enumeration process. This book discusses as well the difficulties that arise with multiplication and inverting programs, and of some ways to avoid or overcome them. The final chapter deals with the computational problems related to invariant factors in linear algebra. Mathematicians as well as students of algebra will find this book useful.

Autoren/Hrsg.

Leech, John

Inhaltsverzeichnis

1;Front Cover;1
2;Computational Problems in Abstract Algebra;4
3;Copyright Page;5
4;Table of Contents;6
5;Foreword;8
6;Preface;10
7;Chapter 1. Investigations of groups on computers;12
7.1;1. Introduction;12
7.2;2. Special purpose programmes;12
7.3;3. General purpose programmes;16
7.4;4. Acknowledgements;25
7.5;REFERENCES;25
8;Chapter 2. Coset enumeration;32
8.1;1. Introduction;32
8.2;2. Hand calculation;32
8.3;3. Coincidences;33
8.4;4. Computer implementation;33
8.5;5. First algorithm;34
8.6;6. Second algorithm;35
8.7;7. Computer handling of coincidences;36
8.8;8. Form of data;37
8.9;9. Termination of the process;38
8.10;10. A word problem;38
8.11;11. A worked example;39
8.12;12. An example with coincidences;42
8.13;13. Acknowledgement;46
8.14;REFERENCES;46
9;Chapter 3. Some examples using toset enumeration;48
9.1;Introduction;48
9.2;REFERENCES;52
10;Chapter 4. Defining relations for subgroups of finite index of groups with a finite presentation;54
10.1;REFERENCES;55
11;Chapter 5. Nielsen transformations;56
11.1;REFERENCE;57
12;Chapter 6. Calculation with the elements of a finite group given by generators and defining relations;58
12.1;1. Preliminary remarks;58
12.2;2. Input data;58
12.3;3. Multiplication algorithm;59
12.4;4. Multiplication programme;61
12.5;5. Some extensions planned;66
12.6;REFERENCES;68
13;Chapter 7. On a programme for the determination of the automorphism group of a finite group;70
13.1;REFERENCES;71
14;Chapter 8. A computational method for determining the automorphism group of a finite solvable group;72
14.1;1. Some results of the theory of Sylow systems and general products;72
14.2;2. Determination of A (G) by composition of allowable automorphisms;75
14.3;3. Preliminaries for the construction of A(G);77
14.4;4. Representation of automorphisms in a computer;78
14.5;5. The computational method for the composition of A(G);79
14.6;6. Some aspects of optimization of the program system;83
14.7;7. Some aspects for the extension of the program system to groups witha normal chain of Hall groups;84
14.8;REFERENCES;85
15;Chapter 9. Combinatorial construction by computer of the set of all subgroups of a finite group by composition of partial sets of its subgroups;86
15.1;1. Preliminaries;86
15.2;2. The generation of the partial set of subgroups;87
15.3;3. The method of constructing the set of all subgroups of ;88
15.4;4. Conclusion;92
15.5;REFERENCES;93
16;Chapter 10. A programme for the drawing of lattices;94
16.1;REFERENCES;98
17;Chapter 11. The construction of the character table of a finite group from generators and relations;100
17.1;Introduction;100
17.2;Derivation of a faithful representation;100
17.3;The mapping of an element into its conjugacy class;102
17.4;Derivation of the centre of the group algebra;102
17.5;Construction of the normal subgroup lattice;103
17.6;The numerical characters from the structure constants;103
17.7;Derivation of the algebraic form from the numerical;105
17.8;Results;107
17.9;REFERENCES;111
18;Chapter 12. A programme for the calculation of characters and representations of finite groups;112
18.1;1. Introduction;112
18.2;2. Available programmes for the investigation of finite groups;112
18.3;3. One-dimensional representations;114
18.4;4. Irreducible representations of higher dimensions;114
18.5;5. The numerical part of the programme;119
18.6;6. Experience with the programme;120
18.7;REFERENCES;120
19;Chapter 13. The characters of the Weyl group E8;122
19.1;1. Introduction;122
19.2;2. The classes;124
19.3;3. The decomposition of Kronecker powers;127
19.4;4. Induced permutation characters;128
19.5;5. Kronecker products with the character 8;130
19.6;6. Blocks of defect one;132
19.7;REFERENCES;141
20;Chapter 14. On some applications of group-theoretical programmes to the derivation of the crystal classes of R4;142
20.1;REFERENCES;146
21;Chapter 15. A search for simple groups of order less than one millions;148
21.1;1. Introduction;148
21.2;2. Notation. List of known simple groups in the range;148
21.3;3. Known results used;150
21.4;4. General outline of the search;156
21.5;5. Examples of application of the general theory;161
21.6;6. Construction of a new simple group of order 604,800;172
21.7;REFERENCES;178
22;Chapter 16. Computational methods in the study of permutation groupst;180
22.1;1. Introduction;180
22.2;2. Simply primitive groups;181
22.3;3. Doubly transitive groups;183
22.4;4. A computer program;187
22.5;5. The primitive groups of degree not exceeding 20;188
22.6;REFERENCES;194
23;Chapter 17. An algorithm related to the restricted Burnside group of prime exponent;196
23.1;Introduction;196
23.2;An associated matrix algebra of a Lie algebra;196
23.3;The associated algebra of a finitely generated Lie algebra;196
23.4;REFERENCES;198
24;Chapter 18. A module-theoretic computation related to the Burnside problem;200
24.1;1. Introductory considerations;200
24.2;2. General approach;201
24.3;3. Generating the matrix;202
24.4;4. The R-module;203
24.5;5. The S7-module;205
24.6;6. Bell-ringing;206
24.7;7. Triangularizing the matrix;206
24.8;8. Present status;209
24.9;REFERENCES;209
25;Chapter 19. Some combinatorial and symbol manipulation programs in group theory;210
25.1;Introduction;210
25.2;Subgroup lattices;210
25.3;Investigation of groups of order p6,p > 2.;211
25.4;Hughes' conjecture and commutator calculations;211
25.5;Acknowledgments;214
25.6;REFERENCES;214
26;Chapter 20. The computation of irreducible representations of finite groups of order 2n, n< 6t;216
26.1;1. Introduction;216
26.2;2. Development of the Cayley tables of 2-groups.;217
26.3;3. The calculation of the irreducible representations of 2-groups;221
26.4;4. Current computer programs;225
26.5;REFERENCES;226
27;Chapter 21. Some examples of man—machine interaction in the solution of mathematical problems;228
27.1;Example 1. The Sandler group;228
27.2;Example 2. Complete Latin squares;231
27.3;Example 3. Commutators;231
27.4;Concluding remarks;233
27.5;REFERENCES;233
28;Chapter 22. Construction and analysis of non-equivalent finite semigroups;234
28.1;1. Introduction;234
28.2;2. The construction algorithm;234
28.3;3. Some analysis results;237
28.4;REFERENCES;238
29;Chapter 23. Some contributions of computation to semigroups and groupoidst;240
29.1;PART I. GROUPOIDS AND THEIR AUTOMORPHISM GROUPS;241
29.1.1;1.1. Introduction;241
29.1.2;1.2. Outline of the proof of Theorem ;241
29.1.3;1.3. Preparation;242
29.1.4;1.4. Groupoids of order _< 3;244
29.1.5;1.5. Groupoids of order 4;249
29.2;PART II. SYSTEM OF OPERATIONS AND EXTENSION THEORY;258
29.2.1;2.1. Introduction;258
29.2.2;2.2. The system of orerations;258
29.2.3;2.3. General product.;259
29.2.4;2.4. Left general product;261
29.2.5;2.5. Sub-general product;262
29.2.6;2.6. Construction of some general products;263
29.3;Acknowledgemen;271
29.4;REFERENCES;271
30;Chapter 24. Simple Word Problems in Universal Algebrast;274
30.1;Introduction;274
30.2;1. Words;275
30.3;2. Substitutions;278
30.4;3. The word problem;282
30.5;4. The completeness theorem.;283
30.6;5. The superposition process;284
30.7;6. Extension to a complete set;287
30.8;7. Computational experiments.;288
30.9;8. Conclusions;307
30.10;REFERENCES;308
31;Chapter 25. The application of computers to research innon-associative algebras;310
31.1;1. Introduction;310
31.2;2. Examples of computer assisted research;310
31.3;3. Jordan algebra identities;312
31.4;REFERENCES;316
32;Chapter 26. Identities in Jordan algebras;318
32.1;REFERENCES;324
33;Chapter 27. On property D neofields and some problems concerning orthogonal latin squares;326
33.1;REFERENCES;330
34;Chapter 28. A projective configuration;332
34.1;REFERENCES;334
35;Chapter 29. The uses of computers in Galois theory;336
35.1;REFERENCES;339
36;Chapter 30. An enumeration of knots and links, and some of their algebraic properties;340
36.1;Introduction;340
36.2;1. Notation for tangles;341
36.3;2. Notation for knots;343
36.4;3. Some tangle equivalences. Flyping;344
36.5;4. Equivalences for knots;346
36.6;5. Orientation and string-labelling;347
36.7;6. Polynomials and potentials;348
36.8;7. Determinants and signature;351
36.9;8. Slice knots and the cobordism group;351
36.10;9. Notes on the tables. Acknowledgments;352
36.11;REFERENCES;369
37;Chapter 31. Computations in knot theory;370
37.1;1. Computer representation of knots;370
37.2;2. Computation of algebraic invariants of knots;371
37.3;3. Manipulation of knot diagrams;373
37.4;REFERENCES;375
38;Chapter 32. Computer experiments on sequences which formintegral bases;376
38.1;REFERENCES;381
39;Chapter 33. Application of computer to algebraic topology on some bicomplex manifoldst;382
39.1;1. Introductory remarks;382
39.2;2. Modular group in one variabl;382
39.3;3. Hilbert modular group;384
39.4;4. Assembling the floor;386
39.5;5. Approximate topological configuration;392
39.6;REFERENCES;392
40;Chapter 34. A real root calculus;394
40.1;REFERENCES;403
41;Chapter 35. Some computational problems and methods related to invariant factors and control theoryt;404
41.1;1. Introduction;404
41.2;2. Common factors of polynomials;405
41.3;3. Theory of realizations;406
41.4;REFERENCES;409
42;List of participants;410