Group Rings of Finite Groups Over p-adic Integers
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1026 Wilhelm Plesken
Group Rings of Finite Groups Over p-adic Integers
Spri nger-Verlag Berlin Heidelberg New York Tokyo 1983
Author
Wilhelm Pies ken Lehrstuhl 0 fOr Mathematik, RWTH Aachen Templergraben 64, 5100 Aachen, Federal Republic of Germany
AMS Subject Classifications (1980): 16A18, 16A26, 16A64, 20C05, 20Cll,20C20 ISBN 3-540-12728-3 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12728-3 Springer-Verlag New York Heidelberg Berlin Tokyo Library of Congress Cataloging in Publication Data. Plesken, Wilhelm, 1950- Group rings of finite groups over p-adic integers. (Lecture notes in mathematics; 1026) Bibliography: p. Includes index. 1. Group rings. 2. Finite groups. 3. p-adic numbers. I. Title. II. Series: Lecture notes in mathematics. (Springer-Verlag); 1026. OA3.L28 no. 1026 [OAI71] 510s [512'.22] 83-16985 ISBN 0-387-12728-3 (U.S.) 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 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
In the present notes the theory of orders over Dedekind domains is applied to study group rings of finite groups over the p-adic integers. The presentation grew out of my Habilitationsschrift at the RheinischWestfalische Technische Hochschule Aachen, but goes far beyond it. The major part of the material is accessible to anyone who knows the definition of a maximal order and is familiar with the elements of modular representation theory of finite groups. It was Professor H. Zassenhaus who introduced me to the subject a couple of years ago, and it is fair to say that these notes would have never been written without him. I acknowledge with pleasure that I greatly profited from discussions with H. Benz, H. Jacobinski, G. Michler, H. Pahlings, and K. Roggenkamp. I am very grateful to W. Rump for reading the manuskript and to a referee for pointing out an error in an earlier version, the correction of which lead to generalizations of some results. For typing the manuscript I would like to thank Mrs. D. Burkel, Miss A. Nijenhuis, and Mrs. C. Schneider. Part of the work was done while I held a Heisenberg scholarship. I would like to thank the Deutsche Forschungsgemeinschaft for the opportunities given to me by this grant. Finally I would like to thank the Sonderforschungsbereich fUr theoretische Mathematik, Bonn, for their hospitality during part of the preparation of the manuscript.
W. Plesken
CONTENTS I. II.
Introduction Graduated and graduable orders a. Definition and characterization of graduated orders
III.
7
b. Graduable orders
21
c. Some properties of graduated orders, graduated h
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