Cohomology of Finite Groups
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Editors
M. Artin S. S. Chern 1. Coates 1. M. Fröhlich H. Hironaka F. Hirzebruch L. Hörmander C. C. Moore 1. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai 1. Tits M. Waldschmidt S.Watanabe Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Alejandro Adern
R. James Milgram
Cohomology of Finite Groups
Springer-Verlag Berlin Heidelberg GmbH
Alejandro Adern Department of Mathernatics University of Wisconsin Madison, Wl 53706, USA R. James Milgram Departrnent of Applied Hornotopy Stanford University Stanford, CA 94305-9701, USA
Mathernatics Subject Classification (1991): 20J05, 20J06, 20110, 55R35, 55R40, 57S17, 18GlO, 18G15, 18G20, 18G40
ISBN 978-3-662-06284-5 ISBN 978-3-662-06282-1 (eBook) DOI 10.1007/978-3-662-06282-1
Library of Congress Cataloging-in-Publication Data Adern, Alejandro. Cohomology of finite groupsl Alejandro Adern, Richard James Milgram. p. cm. - (Grundlehren der mathematischen Wissenschaften; 309) Inciudes bibliographical references and index. 1. Finite groups. 2. Homology theory. I. Milgram, R. James. 11. Title. 111. Series. QA177.A34 1995 512'.55-dc20 94-13318 CIP This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concemed, specifically the rights oftranslation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Originally published by Springer-Verlag Berlin Heidelberg New York in 1994. Softcover reprint of the hardcover 1st edition 1994
Typesetting: Camera-ready copy produced by the authors' output file using aSpringer TEX macro package 41/3140-54321 0 Printed on acid-free paper SPIN 10078665
Table of Contents
Introduction ................................................
1
Chapter I. Group Extensions, Simple Aigebras and Cohomology
o. 1. 2.
3. 4. 5. 6. 7.
8.
Introduction .............................................. Group Extensions ......................................... Extensions Associated to the Quaternions .................... The Group of Unit Quaternions and SO(3) ................... The Generalized Quaternion Groups and Binary Tetrahedral Group ................................................... Central Extensions and SI Bundles on the Torus T 2 ••..••.•.•• The Pull-back Construction and Extensions .................. The Obstruction to Extension When the Center Is Non-Trivial .. Counting the Number of Extensions ......................... The Relation Satisfied by JL(gI, g2, g3) ....................... A Certain Universal Extension .............................. Each Element in H~(G; C) Represents an Obstruction ......... Associative Aigebras and H~(G; C) ..........................
