An introduction to the theory of groups:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1999
|
| Ausgabe: | 4. ed., corr. 2. print. |
| Schriftenreihe: | Graduate texts in mathematics
148 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Bis 2. ed. u.d.T.: Rotman, Joseph J.: The theory of groups |
| Beschreibung: | XIV, 513 S. graph. Darst. |
| ISBN: | 9780387942858 0387942858 3540942858 |
Internformat
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| 245 | 1 | 0 | |a An introduction to the theory of groups |c Joseph J. Rotman |
| 250 | |a 4. ed., corr. 2. print. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1999 | |
| 300 | |a XIV, 513 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 148 | |
| 500 | |a Bis 2. ed. u.d.T.: Rotman, Joseph J.: The theory of groups | ||
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Datensatz im Suchindex
| _version_ | 1804138798021345280 |
|---|---|
| adam_text | Contents
Preface
to the Fourth Edition
vii
From Preface to the Third Edition
ix
To the Reader
xv
Chapter
1
Groups and Homomorphisms
1
Permutations
2
Cycles
3
Factorization into Disjoint Cycles
6
Even and Odd Permutations
7
Semigroups
10
Groups
12
Homomorphisms
16
Chapter
2
The Isomorphism Theorems ~°
Subgroups
Lagrange s Theorem
Cyclic Groups
Normal Subgroups
Quotient Groups
The Isomorphism Theorems
Correspondence Theorem
Direct Products
xii Contents
Chapter
3
Symmetric Groups and G-Sets
43
Conjugates
43
Symmetric Groups
46
The Simplicity of An
50
Some Representation Theorems
51
G-Sets
55
Counting Orbits
58
Some Geometry
63
Chapter
4
The Sylow Theorems
73
p-Groups
73
The Sylow Theorems
78
Groups of Small Order
82
Chapter
5
Normal Series
89
Some Galois Theory
91
The Jordan- Holder Theorem
98
Solvable Groups
102
Two Theorems of P. Hall
108
Central Series and Nilpotent Groups
112
p-Groups
í
19
Chapter
6
Finite Direct Products
125
The Basis Theorem
125
The Fundamental Theorem of Finite Abelian Groups
131
Canonical Forms; Existence
133
Canonical Forms; Uniqueness
141
The Krull-Schmidt Theorem
144
Operator Groups
151
Chapter
7
Extensions and Cohomology
154
The Extension Problem
154
Automorphism Groups
156
Semidirect Products
167
Wreath Products
172
Factor Sets
178
Theorems of Schur-Zassenhaus and
Gaschütz 188
Transfer and Burnside s Theorem
193
Projective Representations and the
Schur
Multiplier
201
Derivations
211
Contents xiii
Chapter
8
Some Simple Linear Groups
217
Finite Fields
217
The General Linear Group
219
PSL(2, K)
224
PSL(m, K)
227
Classical Groups
234
Chapter
9
Permutations and the
Mathieu
Groups
247
Multiple Transitivity
247
Primitive G-Sets
256
Simplicity Criteria
259
Affine
Geometry
264
Projective
Geometry
272
Sharply 3-Transitive Groups
281
Mathieu
Groups
286
Steiner
Systems
293
Chapter
10
Abelian Groups
307
Basics
307
Free Abelian Groups
312
Finitely Generated Abelian Groups
318
Divisible and Reduced Groups
320
Torsion Groups
325
Subgroups of
Q
331
Character Groups
335
Chapter
11
Free Groups and Free Products
343
Generators and Relations
343
Semigroup Interlude
349
Coset Enumeration
351
Presentations and the
Schur
Multiplier
358
Fundamental Groups of Complexes
366
Tietze s Theorem
374
Covering Complexes
377
The Nrielsen
Schreier
Theorem
383
Free Products
388
The Kurosh Theorem
391
The van
Kampen
Theorem
394
Amalgams
401
HNN Extensions
xjv
Contents
Chapter
12
The Word Problem
418
Introduction
418
Turing Machines
420
The Markov-Post Theorem
425
The Novikov-Boone-Britton Theorem: Sufficiency of Boone s
Lemma
430
Cancellation Diagrams
433
The Novikov-Boone-Britton Theorem: Necessity of Boone s
Lemma
438
The Higman Imbedding Theorem
450
Some Applications
464
Epilogue
471
Apphndix I
Some Major Algebraic Systems
475
Appendix II
Equivalence Relations and Equivalence Classes
477
Appendix III
Functions
479
Appendix IV
Zorn s Lemma
481
Appendix V
Countability 4g3
Appendix VI
Commutative Rings 4g5
Bibliography 495
Notation 49g
Index 5O3
Anyone who has studied abstract algebra and linear algebra as an
undergraduate can understand this book. This edition has been com¬
pletely revised and reorganized, without however losing any of the
clarity of presentation that was the hallmark of the previous editions.
The first six chapters provide ample material for a first course: begin¬
ning with the basic properties of groups and homomorphisms, the
topics covered include Lagrange s theorem, the Noether isomorphism
theorems, symmetric groups, G-sets, the Sylow theorems, finite abelian
groups, Krull-Schmidt theorem, solvable and
nilpotent
groups, and the
Jordan-Holder theorem.
The middle portion of the book then uses the Jordan-Holder theorem
to organize the discussion of extensions (automorphism groups, semi-
direct products, the Schur-Zassenhaus lemma,
Schur
multipliers) and
simple groups (simplicity of
projective
unimodular groups and, after a
return to G-sets, a construction of the sporadic
Mathieu
groups).
The book closes with three chapters on: infinite abelian groups, with
emphasis on countable groups; free groups and presentations of
groups, coset enumeration, free products, amalgams, and HNN exten¬
sions; and a complete proof of the unsolvability of the word problem
for finitely presented groups, as well as the Higman imbedding theo¬
rem and the undecidability of the isomorphism problem for groups.
|
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| author | Rotman, Joseph J. 1934- |
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| classification_tum | MAT 200f |
| ctrlnum | (OCoLC)884365713 (DE-599)BVBBV023747553 |
| discipline | Mathematik |
| edition | 4. ed., corr. 2. print. |
| format | Book |
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| indexdate | 2024-07-09T21:33:54Z |
| institution | BVB |
| isbn | 9780387942858 0387942858 3540942858 |
| language | English |
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| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Rotman, Joseph J. 1934- Verfasser (DE-588)120676826 aut An introduction to the theory of groups Joseph J. Rotman 4. ed., corr. 2. print. New York [u.a.] Springer 1999 XIV, 513 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 148 Bis 2. ed. u.d.T.: Rotman, Joseph J.: The theory of groups Gruppentheorie Gruppe Mathematik (DE-588)4022379-6 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Gruppe Mathematik (DE-588)4022379-6 s DE-604 Gruppentheorie (DE-588)4072157-7 s Graduate texts in mathematics 148 (DE-604)BV000000067 148 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017296628&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017296628&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Rotman, Joseph J. 1934- An introduction to the theory of groups Graduate texts in mathematics Gruppentheorie Gruppe Mathematik (DE-588)4022379-6 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| subject_GND | (DE-588)4022379-6 (DE-588)4072157-7 (DE-588)4151278-9 |
| title | An introduction to the theory of groups |
| title_auth | An introduction to the theory of groups |
| title_exact_search | An introduction to the theory of groups |
| title_full | An introduction to the theory of groups Joseph J. Rotman |
| title_fullStr | An introduction to the theory of groups Joseph J. Rotman |
| title_full_unstemmed | An introduction to the theory of groups Joseph J. Rotman |
| title_short | An introduction to the theory of groups |
| title_sort | an introduction to the theory of groups |
| topic | Gruppentheorie Gruppe Mathematik (DE-588)4022379-6 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| topic_facet | Gruppentheorie Gruppe Mathematik Einführung |
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