Lecture Notes in Mathematics

Transcription

1 Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 715 Inder Bir S. Passi Group Rings and Their Augmentation Ideals Springer-Verlag Berlin Heidelberg New York 1979

2 Author Inder Bir S. Passi Department of Mathematics Kurukshetra University Kurukshetra (Haryana) India AMS Subject Classifications (1970): 16A26, 20C05 ISBN Springer-Verlag Berlin Heidelberg NewYork ISBN Springer-Verlag NewYork 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 the publisher, the amount of the fee to be determined by agreement with the publishel by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/

3 INTRODUCTION The purpose of these Notes is to report on some aspects of the augmentation ideal of a group ring. Every two-sided ideal in a group ring determines a normal subgroup. One of the oldest and most challenging problems in group rings is the identification of the normal subgroups, called dimension subgroups, determined by the powers of the augmentation ideal of a group ring. The first five Chapters of these Notes are devoted to this problem. Dimension subgroups were first studied by Magnus [48] who proved that, for free groups, the integral dimension subgroup series coincides with the lower central series. More than thirty years later Rips [78] demonstrated that Magnus' theorem cannot be generalized to arbitrary groups. Sandling ([79], [80]) initiated the study of dimension subgroups over arbitrary coefficient rings and of Lie dimension subgroups. Both the associative and the Lie powers [79] of an augmentation ideal are polynomial ideals [59]. We, therefore, begin in Chapter I with the study of polynomial ideals. The main result proved in Chapter II is that for the study of (Lie) dimension subgroups over arbitrary coefficient rings it is enough to restrict to integral and modular coefficients. Chapter III is an exposition of a portion of a fundamental paper of Hartley [27]. Our aim here is to show that Hartley's theory provides a frame work in which it is possible to unify the approaches of Jennings ([35], [36]) and Lazard [38] for the study of augmentation powers. Dimension subgroups over fields have been studied by Bovdi [7], Hall [26], Jennings ([35], [36]), Lazard [38], Sandling [80] and Zassenhaus [99]. We calculate these subgroups in Chapter IV. We study in Chapter V polynomial maps on groups and calculate integral dimension subgroups in certain cases. At the present time, apart from some special classes of groups, complete description of only the first five integral dimension subgroups is known and the general problem remains as challenging as ever. Chapters VI and VII are devoted to the study of the intersection of the powers of an augmentation ideal. Lichtman [39] has recently characterized group theoretically the residual nilpotence of the augmentation ideal (over integers). A reasonable description of the intersection of the powers of an augmentation ideal, however, is still not known. The associated graded ring arising from the filtration of a group

4 IV ring given by the powers of its augmentation ideal is the subject of study in Chapter VIII. This is an object which is extremely rich in algebraic structure. As for dimension subgroups, the structure of the associated graded ring can be nicely described when the coefficients are in a field (Quillen [77]). However, the general case remains little understood. I would like to express my sincere thanks to my teacher D. Rees who introduced me to group rings. I am indebted to Brian Hartley and Robert Sandling for critically reading parts of these Notes and making several helpful comments. The writing of these Notes was undertaken during the year when I was visiting the University of Alberta. The completion of these Notes is the result of my stay at the University of Warwick during the 1978 Symposium on Infinite Groups and Group Rings. The hospitality of the University of Alberta and the University of Warwick is gratefully acknowledged. I am thankful to my colleagues J.N. Mital and L.R. Vermani and research students M. Goyal and S.K. Arora for numerous stimulating discussions. Thanks are also due to K.W. Gruenberg, N.D. Gupta, D.S. Passman and U. Stammbach for their interest in these Notes and the encouragement they have given me. September, 1978 Kurukshetra University Kurukshetra (Haryana) India Current address: Centre for Advanced Study in Mathematics Panjab University Chandigarh India

5 CONTENTS Chapter I: Polynomial Ideals in Group Rings... I I. Polynomial ideals... I Chapter II: Dimension Subgroups I. Subgroups determined by polynomial ideals Dimension subgroups over arbitrary rings of coefficients Lie dimension subgroups over arbitrary rings of coefficients Chapter III: Group Rings of Nilpotent Groups I. N-series and filtrations of the augmentation ideal A construction in group rings of nilpotent groups Chapter IV: Dimension Subgroups over Fields I. Dimension subgroups over fields Lie dimension subgroups over fields Applications Chapter V: Polynomial Maps on Groups and Dimension Subgroups 50 I. Polynomial maps A characterization of polynomial maps on groups Wreath product Relationship with dimension subgroups Integral dimension subgroups The fourth integral dimension subgroup Modular dimension subgroups Chapter VI: The Residual Nilpotence of the Augmentation Ideal 76 I. The nilpotence of the augmentation ideal The residual nilpotence of the augmentation ideal Chapter VII: The Nilpotent Residue of an Augmentation Ideal I. Elements of infinite p-height A~(G) in periodic groups z ~(G) 3- A R in finitely generated nilpotent groups ~ (G) over fields The intersection theorem

6 VI Chapter VIII: The Associated Graded Ring of a Group Ring The associated graded ring of a group ring The graded Lie ring associated to an N-series The Lie ring ~(G) The Lie algebra SF (G) P 5- The associated graded algebra of a group algebra The associated graded ring of an integral group ring The periodicity of the groups Qu(G) The homomorphism e n References: