Fundamentals of Group Theory An Advanced Approach
Fundamentals of Group Theory provides an advanced look at the basic theory of groups. Standard topics in the field are covered alongside a great deal of unique content. There is an emphasis on universality when discussing the isomorphism theorems, quotien
- PDF / 4,053,979 Bytes
- 385 Pages / 439.37 x 666.142 pts Page_size
- 151 Downloads / 1,061 Views
Fundamentals of Group Theory An Advanced Approach
Steven Roman Irvine, CA USA
ISBN 978-0-8176-8300-9 e-ISBN 978-0-8176-8301-6 DOI 10.1007/978-0-8176-8301-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941115 Mathematics Subject Classification (2010): 20-01 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper
Birkhäuser Boston is part of Springer Science+Business Media (www.birkhauser.com)
To Donna
Preface
This book is intended to be an advanced look at the basic theory of groups, suitable for a graduate class in group theory, part of a graduate class in abstract algebra or for independent study. It can also be read by advanced undergraduates. Indeed, I assume no specific background in group theory, but do assume some level of mathematical sophistication on the part of the reader. A look at the table of contents will reveal that the overall topic selection is more or less standard for a book on this subject. Let me at least mention a few of the perhaps less standard topics covered in the book: 1) An historical look at how Galois viewed groups. 2) The problem of whether the commutator subgroup of a group is the same as the set of commutators of the group, including an example of when this is not the case. 3) A discussion of xY-groups, in particular, a) groups in which all subgroups have a complement b) groups in which all normal subgroups have a complement c) groups in which all subgroups are direct summands d) groups in which all normal subgroups are direct summands. 4) The subnormal join property, that is, the property that the join of two subnormal subgroups is subnormal. 5) Cancellation in direct sums: A group K is cancellable in direct sums if E { K ¸ F { Lß
K¸L
Ê
E¸F
(The symbol { represents the external direct sum.) We include a proof that any finite group is cancellable in direct sums. 6) A complete proof of the theorem of Baer that a nonabelian group K has the property that all of its subgroups are normal if and only if KœUEF where U is a quaternion group, E is an elementary abelian group of exponent # and F is an abelian group all of whose elements have odd order.
vii
viii
Preface
7) A somewhat more in-depth discussion of the structure of :-groups, including the nature of conjugates in a :-gr
Data Loading...
