Almost Completely Decomposable Groups

366 Pages·2000·30.318 MB·\366

by  A Mader (Author)

ALMOST COMPLETELY
DECOMPOSABLE GROUPS
ALGEBRA, LOGIC AND APPLICATIONS
A series edited by
R. Gobel
Universitat Gesamthochschule, Essen, Germany
A. Macintyre
University of Edinburgh, UK
Volume 1
Linear Algebra and Geometry
A.I. Kostrikin and Yu.I. Manin
Volume 2
Model Theoretic Algebra: With Particular Emphasis on Fields, Rings, Modules
Christian U. Jensen and Helmut Lenzing
Volume 3
Foundations of Module and Ring Theory: A Handbook for Study and Research
Robert Wisbauer
Volume 4
Linear Representations of Partially Ordered Sets and Vector Space Categories
Daniel Sims on
Volume 5
Semantics of Programming Languages and Model Theory
M. Droste and Y Gurevich
Volume 6
Exercises in Algebra: A Collection of Exercises in Algebra, Linear Algebra and
Geometry
Edited by A.I. Kostrikin
Volume 7
Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms
Kazimierz Szymiczek
Volume 8
Multilinear Algebra
Russell Merris
Volume 9
Advances in Algebra and Model Theory
Edited by Manfred Droste and Rudiger Gobel
Please see the back of this book for other titles in the Algebra, Logic and
Applications series.
ALMOST COMPLETELY
DECOMPOSABLE GROUPS
Adolf Mader
University of Hawaii, USA
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
First published 2000 by Gordon and Breach Science Publishers
Published 2018 by CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2000 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
ISBN 13: 978-90-5699-225-5 (hbk)
This book contains information obtained from authentic and highly
regarded sources. Reasonable efforts have been made to publish reliable
data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their
use. The authors and publishers have attempted to trace the copyright
holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been
obtained. If any copyright material has not been acknowledged please write
and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be
reprinted, reproduced, transmitted, or utilized in any form by any
electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any
information storage or retrieval system, without written permission from
the publishers.
For permission to photocopy or use material electronically from this work,
please accesswww.copyright.com(http://www.copyright.com/) or contact
the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive,
Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization
that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a
separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or
registered trademarks, and are used only for identification and explanation
without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http:/ /www.crcpress.com
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library.
ISSN: 1041-5394
Rosa Mader (1907—1995) gewidmet

Contents
Introduction 1
Acknowledgements 4
1. Notation and Background 5
1.1. Generalities 5
1.2. Rational Groups 6
1.3. The Anatomy of Finite Posets 13
1.4. Unit Groups 17
1.5. Integal Linear Algebra 19
1.6. Exercises 23
1.7. Solutions to Exercises 25
2. Basics and Completely Decomposable Groups 27
2.1. Rank, Coefficient Groups, and Purity 27
2.2. Rank-One Groups and Types 30
2.3. Socles and Radicals 32
2.4. Completely Decomposable Groups 36
2.5. Bounded Quotients of Completely Decomposable Groups 43
2.6. Homocyclic Finite Groups 49
2.7. Exercises 51
2.8. Questions 57
2.9. Solutions to Exercises 57
3. Cyclic Essential Extensions 63
3.1. Cyclic Essential Extensions 63
3.2. Examples 65
3.3. Exercises 70
4. Regulating Subgroups and Regulators 73
4.1. Butler Groups 74
vii
viii CONTENTS
4.2. Regulating Subgroups 78
4.3. Counting Regulating Subgroups 85
4.4. The Regulator 87
4.5. Odds and Ends 93
4.6. Exercises 98
4.7. Questions 100
4.8. Solutions to Exercises 101
5. Local-Global Relationships 103
5.1. Primary Constituents 103
5.2. Localization of the Regulator Criterion 106
5.3. Regulating Subgroups of Primary Constituents 107
5.4. Questions 108
6. Groups with Cyclic Regulating Quotient 109
6.1. Notation and Basics 110
6.2. Regulating Subgoups 111
6.3. Clipped crq-Groups 114
6.4. Local crq-Groups 118
6.5. Direct Decomposition of Local crq-Groups 119
6.6. Regulating Quotients of Local crq-Groups 127
6.7. Exercises 133
6.8. Questions 136
6.9. Solutions to Exercises 136
7. Completely Decomposable Summands 141
7.1. Width and Rank 141
7.2. Radicals and Co-Butler Decompositions 143
7.3. Completely Decomposable Summands and Clipped
Groups 146
7.4. Exercises 151
7.5. Questions 152
7.6. Solutions to Exercises 153
8. Anti-Representations 155
8.1. Anti-Representations 15 6
8.2. Isomorphism 161
8.3. Exercises 169
8.4. Solutions to Exercises 170
9. Near-Isomorphism and Type-Isomorphism 171
9.1 Near-Isomorphism 171
CONTENTS ix
9.2. Type-Isomorphism is Near-Isomorphism 176
9.3. Exercises 181
9.4. Questions 181
9.5. Solutions to Exercises 182
10. Fundamental Decomposition Theorems 185
10.1. Lifting Idempotents 186
10.2. Arnold’s Theorem 189
10.3. The Azumaya Theorem 192
10.4. The Faticoni-Schultz Theorem 195
10.5. Exercises 197
10.6. Questions 198
10.7. Solutions to Exercises 198
11. Finite Essential Extensions 199
11.1. Standard Description 199
11.2. Greatest Common Divisors 202
11.3. Computation of Greatest Common Divisors 208
11.4. Purification of Base Group Summands 210
11.5. Burkhardt Invariants 212
11.6. Primary Constituents 214
11.7. Representing Matrices 215
11.8. Two Existence Theorems 216
11.9. Completely Decomposable Direct Summands 218
11.10. A Big Example and Maple Procedures 221
11.11. Some Additional Maple Procedures 237
11.12. Exercises 238
11.13. Questions 242
11.14. Solutions to Exercises 242
12. Classification 249
12.1. crq-Group 249
12.2. Near-Isomorphism Invariants 252
12.3. Two Critical Types 254
12.4. Homocyclic Regulator Quotients 257
12.5. Uniform Groups 259
12.6. Global Block-Rigid crq-Groups and Isomorphism 264
12.7. Some Applications 269
12.8. Exercises 272
12.9. Questions 276
12.10. Solutions to Exercises 276