Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch:
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1972
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| 100 | 1 | |a Plotkin, Boris Isaakovič |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch |
| 264 | 1 | |c 1972 | |
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 505 | 0 | |a Groningen: Wolters - Noordhoff 1972. XVIII, 502 S.<br>Übers. aus dem Russ. | |
| 700 | 1 | |a Hirsch, K. A. |e Sonstige |4 oth | |
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Record in the Search Index
| _version_ | 1804130982846005248 |
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| adam_text | TABLE OF CONTENTS
Preface xm
I General Problems
Chapter 1
Algebraic systems
1.1 Algebras 1
1.1.1 Definitions, 1
1.1.2 Congruences and homomorphisms, 5
1.1.3 Direct and subdirect sums of algebras, 9
1.1.4 Free and reduced free algebras, 10
1.1.5 Commutative algebras. The commutator algebra, 13
1.1.6 Quasi-endomorphisms of algebras, ^-semigroups, 14
1.2 Multioperator groups 18
1.2.1 Definitions, 18
1.2.2 Near-rings. Multioperator near-rings, 20
1.2.3 Normal series and systems, 23
1.2.4 Abelianness, nilpotency, solubility, 24
1.2.5 Direct decompositions. Complete reducibility, 26
1.2.6 Radicals in multioperator groups, 31
1.3 Models. General algebraic systems 36
1.3.1 Definitions, 36
1.3.2 The language of the lower predicate calculus, 40
1.3.3 Direct products of models. Ultraproducts, 44
1.3.4 The local theorem of Godel-Mal tsev, 48
1.3.5 Multi-based algebraic systems, 53
1.3.6 Concluding remarks, 54
Chapter 2
Groups of automorphisms of algebraic systems
2.1 Representations and their associated pairs 58
2.1.1 Preliminary remarks, 58
2.1.2 Homomorphisms of pairs. Subpairs, 60
V
2.1.3 Transitive pairs, 67
2.1.4 Direct products of pairs and other constructions, 72
2.1.5 Radicals connected with representations, 77
2.2 Existence theorems for representations 79
2.2.1 Preliminary remarks, 79
2.2.2 The theorem of Mostowski-Ehrenfeucht, 81
2.2.3 A theorem of Birkhoff, 86
2.2.4 Local theorems on representations, 87
2.2.5 Miscellany, 91
2.3 Galois connections 93
2.3.1 Galois connections for pairs, 93
2.3.2 Extension of representations. The Bourbaki scale, 95
2.3.3 More on Galois connections, 98
2.4 Some outer properties and radicals 101
2.4.1 A method of forming outer properties, 101
2.4.2 Radicals of the acting group connected with distinguished properties, 104
2.4.3 Almost periodicity and relative finiteness, 106
2.4.4 Relative nilpotency and relative solubility, 108
2.4.5 An example, 111
2.4.6 Boundedness and algebraicity, 112
2.5 Representations of groups and of {2-semigroups 114
2.5.1 Representations of fl-semigroups and generalized modules 114
2.5.2 fi-semigroups associated with semigroups (or groups), 117
2.5.3 Multioperator near-rings and representations on J2-groups, 120
2.5.4 More on boundedness and algebraicity, 123
Chapter 3
Automorphisms groups of multioperator groups
3.1 Reducibility and irreducibility 127
3.1.1 Reducibility, 127
3.1.2 Irreducibility, 133
3.1.3 The radical of a representation, 135
3.1.4 The Jacobson radical of an f2-near-ring, 140
3.1.5 Generalizations, 141
VI
3.2 Decomposability, complete reducibility, imprimitivity . . . 142
3.2.1 Preliminary remarks. Maschke s theorem, 142
3.2.2 Clifford s theorem, 147
3.2.3 Imprimitivity and primitivity, 149
3.2.4 The group of all automorphisms of a completely reducible 0-group, 153
3.3 Stability and associated radicals 153
3.3.1 Finite stability. The y-radical, 153
3.3.2 A remark on local boundedness of a representation, 157
3.3.3 Quasistability. The j8-radical, 159
3.3.4 On nilsets in the acting group, 161
3.3.5 Relations between the three radicals, 164
3.3.6 Radicals of the domain of action, 167
3.3.7 General triangularity, 168
3.3.8 Invertibilityofanendomorphismof an Q-group, 170
3.4 Additional remarks on generalized modules 173
3.4.1 Annihilator properties, 173
3.4.2 Annihilator properties and nil-properties, 177
3.4.3 The Levitzki radical of an £2-near-ring, 180
3.5 Automorphism groups of direct sums of £2-groups . . . . 182
3.5.1 General remarks, 182
3.5.2 Triangular groups of automorphisms connected with direct decomposi¬
tions, 183
3.5.3 Generalized matrices, 186
Chapter 4
Groups of automorphisms of vector spaces
4.1 Linear representations and linear pairs 191
4.1.1 Introduction, 191
4.1.2 Matrices and matrix representations, 195
4.1.3 Triangularity and stability in linear pairs, 202
4.1.4 Finiteness conditions, 206
4.2 Finite-dimensional linear representations 209
4.2.1 Introductory remarks, 209
Vll
4.2.2 The local theorem, 211
4.2.3 Finitely generated groups, 213
4.2.4 Characters, 219
4.3 Some properties of infinite-dimensional linear groups . . . 226
4.3.1 Algebraic elements and local finite-dimensionality, 226
4.3.2 Radicals of a linear group and radicals of its hull, 230
4.4 Groups of automorphisms of linear algebraic systems . . . 235
4.4.1 Linear algebraic systems, 235
4.4.2 Algebraic linear groups, 236
4.4.3 Regular automorphisms of linear systems, 243
4.4.4 Miscellany, 248
4.5 The full linear group and the full symmetric group . . . 252
4.5.1 The full symmetric group, 252
4.5.2 The full monomial group, 259
4.5.3 The full linear group. A survey, 260
VIII
II Automorphism groups of groups
Chapter 5
Some information from the abstract theory of groups
5.1 Group pairs 265
5.1.1 Group pairs and semidirect products, 265
5.1.2 The holomorph, 267
5.1.3 Connections with the group of inner automorphisms, 270
5.1.4 Two remarks on reducibility in group pairs, 272
5.2 Radicals in groups 273
5.2.1 Radicals and ascendancy, 273
5.2.2 Some specific radicals, 277
5.2.3 The Baer radical and the Gruenberg radical, 281
5.2.4 Minimal radical classes, 286
5.2.5 Locally bounded groups, 287
5.2.6 The a-radical of a group. Radicals connected with an inner pair, 289
5.3 Radical groups and ^-groups 290
5.3.1 Semisimplicity. The completely reducible radical, 290
5.3.2 Plotkin groups and JP-groups, 292
5.3.3 A theorem on isomorphisms of series, 295
5.3.4 Normal subgroups bounding the group, 298
5.4 Left Engel elements in groups. Examples 302
5.4.1 The Hirsch-Plotkin radical and left Engel elements, 302
5.4.2 Left Engel groups. Right Engel elements, 310
5.4.3 Examples, 312
Chapter 6
Automorphisms of groups
6.1 Types of automorphisms 324
6.1.1 Locally inner automorphisms, 324
6.1.2 Outer automorphisms, 325
6.1.3 Normal and central endomorphisms and automorphisms, 325
IX
6.1.4 Nilautomorphisms, 328
6.1.5 Regular automorphisms, 328
6.1.6 Miscellany, 332
6.2 Complete groups. The automorphism group tower .... 334
6.2.1 Complete groups, 334
6.2.2 Definition of the tower, 336
6.2.3 Wielandt s theorem, 338
6.3 The automorphism group of the holomorph 343
6.3.1 Completeness of the holomorph, 343
6.3.2 The automorphism group of the holomorph, 345
6.4 Further questions 349
6.4.1 More on quasi-rings generated by endomorphisms of a group, 349
6.4.2 Residual flniteness of automorphism groups, 350
Chapter 7
Stability and nilpotency
7.1 Finite stability and nilpotency 353
7.1.1 Introduction, 353
7.1.2 The theorems ofKaluzhnin and Hall, 354
7.1.3 Auxiliary lemmas, 357
7.1.4 Additional remarks, 360
7.2 Quasistable and outer locally nilpotent groups of automorphisms 362
7.2.1 Statement of the problem, 362
7.2.2 Stability and radicals of the domain of action, 364
7.2.3 Relations between outer local nilpotency and quasistability of the acting
group, 368
7.2.4 Examples, 370
7.3 Some properties of quasistable groups 372
7.3.1 Pure automorphisms, 372
7.3.2 Almost periodic automorphisms, 376
7.3.3 The existence of a central system in a quasistable group of
automorphisms, 379
X
7.3.4 A refinement for torsion-free groups, 381
7.3.5 Examples, 383
7.4 Sufficient conditions for stability 386
7.4.1 The derived group and stability, 386
7.4.2 The derived group and quasistability, 388
7.4.3 The role of the centre, 390
7.5 Radicals of stable type in group pairs 392
7.5.1 Radicals of the acting group, 392
7.5.2 Connections with the ct-radical, 398
7.5.3 Radicals in the domain of action, 400
Chapter 8
Finiteness conditions in group pairs
8.1 Stability and finiteness conditions 403
8.1.1 The rank of a group, 403
8.1.2 Finiteness of the rank of the domain of action, 405
8.1.3 Finiteness of the rank of the acting group, 409
8.1.4 Chain conditions in the acting group, 411
8.1.5 Chain conditions in the domain of action, 414
8.1.6 More on relations between the a- and ^-radicals, 415
8.2 The operator case 417
8.2.1 General remarks, 417
8.2.2 Another generalization of Zassenhaus theorem, 420
8.2.3 More on triangularity, 425
Chapter 9
Automorphism groups of soluble and nilpotent groups
9.1 Automorphisms of Abelian groups 428
9.1.1 Introduction, 428
9.1.2 Periodic groups, 428
9.1.3 Torsion-free groups. References, 430
XI
9.2 Periodic, soluble and nilpotent linear groups 431
9.2.1 Periodic matrix groups, 431
9.2.2 Soluble groups, 437
9.2.3 Nilpotent matrix groups, 441
9.2.4 Generalizations to the infinite-dimensional case, 446
9.2.5 Existence of faithful representations, 447
9.3 Groups of automorphisms of soluble groups 448
9.3.1 Initial remarks, 448
9.3.2 Automorphism groups of soluble /^-groups, 449
9.3.3 Complete groups of automorphisms, 454
9.3.4 More on finiteness conditions in groups, 455
9.4 Automorphisms of nilpotent groups 458
9.4.1 Existence of outer automorphisms, 458
9.4.2 Miscellany, 459
9.4.3 Automorphisms of free nilpotent groups, 461
Appendix 464
Bibliography 477
Subject index 496
XII
|
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| author | Plotkin, Boris Isaakovič |
| author_facet | Plotkin, Boris Isaakovič |
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| bvnumber | BV018365677 |
| contents | Groningen: Wolters - Noordhoff 1972. XVIII, 502 S.<br>Übers. aus dem Russ. |
| ctrlnum | (OCoLC)164617996 (DE-599)BVBBV018365677 |
| format | Book |
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| spelling | Plotkin, Boris Isaakovič Verfasser aut Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch 1972 txt rdacontent n rdamedia nc rdacarrier Groningen: Wolters - Noordhoff 1972. XVIII, 502 S.<br>Übers. aus dem Russ. Hirsch, K. A. Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=011136421&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Plotkin, Boris Isaakovič Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch Groningen: Wolters - Noordhoff 1972. XVIII, 502 S.<br>Übers. aus dem Russ. |
| title | Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch |
| title_auth | Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch |
| title_exact_search | Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch |
| title_full | Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch |
| title_fullStr | Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch |
| title_full_unstemmed | Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch |
| title_short | Groups of automorphisms of algebraic systems (Gruppy avtomorfizmov algebraic̆eskich sistem. engl.) Transl. by K.A. Hirsch |
| title_sort | groups of automorphisms of algebraic systems gruppy avtomorfizmov algebraiceskich sistem engl transl by k a hirsch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=011136421&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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