Theory of K-Loops
The book contains the first systematic exposition of the current known theory of K-loops, as well as some new material. In particular, big classes of examples are constructed. The theory for sharply 2-transitive groups is generalized to the theory of Frob
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Hubert Kiechle
Theory of K- Loops
Springer
Author Hubert Kiechle
SP Geometry and Discrete Mathematics University of Hamburg Bundesstrasse SS 20146 Hamburg, Germany e-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Kiechle, Hubert: Theory of K-loops I Hubert Kiechle. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Tokyo: Springer, 2002
(Lecture notes in mathematics ; 1778) ISBN 3-540-43262-0
Mathematics Subject Classification (2000): 20NOS
ISSN 0075-8434 ISBN 3-540-43262-0 Springer-Verlag Berlin Heidelberg NewYork This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, 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 Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002
Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author
SPIN: 10866597
41/3142/ou - 543210 - Printed on acid-free paper
To my beloved wife Maria
Preface This book contains the first systematic exposition of the presently known theory of K-loops. Besides this, it presents some new results and many examples. Furthermore, the two most important applications are described in detail. Since about ten years the subject of K-loops has grown rapidly, so it seemed reasonable to put things in order. There are not many books on quasigroup and loop theory. The oldest are Bruck’s [20] and Belousov’s [10]. Pflugfelder’s [94] is used as a general reference for the collection of survey articles [24]. Some basic loop theory is also contained in books on projective planes, such as Pickert’s [95]. More specialized, but with very different orientation are Sabinin’s recent [107] and Ungar’s most recent [119] publications. Therefore, most of the material covered has not appeared in book form before. Chapters 1–6 try to unfold the theory in a coherent and self contained way, and could be used as a text. The only prerequisite is basic algebra, in particular, group theory. With very few exceptions 1 complete proofs are given. Examples are later given in Chapters 9,11,12. However, for a course C
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