Group Extensions, Representations, and the Schur Multiplicator
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958
F: Rudolf Beyl
JUrgen Tappe
Group Extensions, Representations, and the Schur Multiplicator
Springer-Verlag Berlin Heidelberg New York 1982
Authors
F Rudolf Beyl Mathematisches Institut der Universitat 1m Neuenheimer Feld 288, 6900 Heidelberg, Germany JUrgen Tappe Lehrstuhl fur Mathematik Rhein.-Westf. Technische Hochschule Aachen Templergraben 55, 5100 Aachen, Germany
AMS Subject Classifications (1980): 20C25, 20E22, 20J05, 20C20, 20E10,20J06 ISBN 3-540-11954-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11954-X Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
TABLE OF CONTENTS
Introduction Chapter 1.
1 Group Extensions with Abelian Kernel
1. The Calculus of Induced Extensions
5
2. The Exact Sequence for Opext
19
3. The Schur Multiplicator and the Universal Coefficient Theorem
28
4. The Ganea Map of Central Extensions
40
5. Compatibility with Other Approaches
47
6. Corestriction (Transfer)
58
Chapter II.
Schur's Theory of Projective Representations
1. Projective Representations
67
2. The Problem of Lifting Homomorphisms
77
3. Representation Groups
91
4. Representation Groups of Free and Direct Products
101
5. The Covering Theory of Perfect Groups
113
Chapter III.
Isoclinism
1. Isoclinic Groups and Central Extensions
123
2. Isoclinism and the Schur Multiplicator
137
3. The Isomorphism Classes of Isoclinic Central Extensions and the Hall Formulae
144
4. On Presentations of Isoclinic Groups
155
5. Representations of Isoclinic Groups
169
IV
Chapter IV.
Other Group-Theoretic Applications of the Schur Multiplicator
1. Deficiency of Finitely Presented Groups
179
2. Metacyclic Groups
193
3. The Precise Center of an Extension Group and Capable Groups
204
4. Examples of the Computation of Z*(G)
213
5. Preliminaries on Group Varieties
227
6. Central Extensions and Varieties
233
7. Schur-Baer MUltiplicators and Isologism
244
Bibliography
261
Index of Special Symbols
271
Subject Index
274
INTRODUCTION
The aim of these notes is a unified treatment of various grouptheoretic topics for which, as it turns out, the Schur multiplicator is the key.
At the beginning of this century, classical projective
geometry was at its peak, while representation theory was growing in the hands of Frobenius and Burnside.
In this climate our subject
started with the two important papers of Schur [1],[2] on the projective representations of finite groups. But it was only in the light of the much more rec
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