Nilpotent Groups
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513 Robert B.Warfield, Jr.
Nilpotent Groups mm
Springer-Verlag Berlin.Heidelberg. New York 1976
Author Robert B. Warfield, Jr. Department of Mathematics University of Washington Seattle, Washington 98195 USA
Library of Congress CaCalaglag la PablicaUon Data
Warfield, Robert B Nilpotent groups.
1940-
(Lecture notes in mathematics ; 513) Bibliography: po Includes index. 1. Groups, Nilpotent. I. Title. II. Series: Lec~ r e notes in mathematics (Berlin) ; 513. [O~lT1] 510'.8s [512'.2] 76-7371 ~ 3 . 1 2 8 no. 513
AMS Subject Classifications (1970): 20E15, 20F20, 20F40, 2 0 H 2 5 ISBN 3-540-07683-2 Springer-Verlag Berlin Heidelberg 9 New 9 York ISBN 0-387-07683-2 Springer-Verlag New Y o r k - Heidelberg Berlin 9 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 w 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 publisher. 9 by Springer-Verlag Berlin Heidelberg 9 1976 Printed in Germany Printing and binding: 8eltz Offsetdruck, Hemsbach/Bergstr.
CONTENTS
i,
Rudiments .....................................................
i
2.
The Upper
6
3.
Tensor
4.
Idempotent
5.
Groups
6
The H a l l - P e t r e s c o
?
Completions,
8
Localization .................................................
9
Nilpotent
Actions,
i0
Nilpotent
Groups
ii
Unipotent
Representations
Central
Products
Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
and the L o w e r
Radicals
with
Abelian
Central
on the C a t e g o r y Central
Formula
of N i l p o t e n t
Groups ...... 17
Q u o t i e n t . . . . . . . . . . . . . . . . . . . . . . . . . 26
and R e s i d u a l
and the S t r u c t u r e
Kolchin's
Admitting
Series . . . . . . . . . . . . . . . . . . 9
B o u n d e d n e s s ........... 43
of C o m p l e t e
Theorem,
Exponents and M a l c e v
Groups ............ 52
and Engel
Conditions...77
in a Ring ............... 83 Completions
of R - g r o u p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.
The M a l c e v
61
Correspondence ...................................
96 104
INTRODUCTION
This is an account of some aspects of the theory of nilpotent groups,
c u l m i n a t i n g with a study of p-adlc completions,
the theory of groups a d m i t t i n g e x p o n e n t s on unipotent representations. the l o c a l i z a t i o n of nilpotent on nilpotent Z~rich. This,
groups
localizatlons,
in a ring, and some results
It includes an earlier set of notes on groups
(1972) and some earlier lectures
(1973) which were available
from the E.T.H.
in
An attempt has been made to make these notes self-contalned.
of course, means that there is a c o n s i d e r
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