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Jaisingh Lloyd R - Schaums Outline of Abstract Algebra

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SCHAUMS OUTLINE OF
Theory and Problems of
ABSTRACT ALGEBRASCHAUMS OUTLINE OF
Theory and Problems of

ABSTRACT ALGEBRA
Second EditionFRANK AYRES, Jr., Ph.D
LLOYD R. JAISINGH
Professor of Mathematics
Morehead State University
Schaums Outline Series FRANK AYRES Jr PhD was formerly Professor and Head of the Department of - photo 1 FRANK AYRES, Jr., Ph.D., was formerly Professor and Head of the Department of Mathematics at Dickinson College, Carlisle, Pennsylvania. He is the author or coauthor of eight Schaums Outlines, including Calculus, Trigonometry, Differential Equations, and Modern Abstract Algebra. LLOYD R. JAISINGH is professor of Mathematics at Morehead State University (Kentucky) for the past eighteen years. He has taught mathematics and statistics during that time and has extensively integrated technology into the classroom.

He has developed numerous activities that involve the MINITAB software, the EXCEL software, and the TI-83+ calculator. He was the recipient of the Outstanding Researcher and Teacher of the Year awards at Morehead State University. His most recent publication is the book entitled Statistics for the Utterly Confused, McGraw-Hill publishing. Copyright 2004 1965 by the McGraw-Hill Companies Inc All rights reserved - photo 2 Copyright 2004, 1965 by the McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.

ISBN: 978-0-07-143098-2
MHID: 0-07-143098-9 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-140327-6, MHID: 0-07-140327-2. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (McGraw-Hill) and its licensors reserve all rights in and to the work. (McGraw-Hill) and its licensors reserve all rights in and to the work.

Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hills prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED AS IS. McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.

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PREFACE
This book on algebraic systems is designed to be used either as a supplement to current texts or as a stand-alone text for a course in modern abstract algebra at the junior and/or senior levels.

In addition, graduate students can use this book as a source for review. As such, this book is intended to provide a solid foundation for future study of a variety of systems rather than to be a study in depth of any one or more. The basic ingredients of algebraic systemssets of elements, relations, operations, and mappingsare discussed in the first two chapters. The format established for this book is as follows: a simple and concise presentation of each topic a wide variety of familiar examples proofs of most theorems included among the solved problems a carefully selected set of supplementary exercises In this upgrade, the text has made an effort to use standard notations for the set of natural numbers, the set of integers, the set of rational numbers, and the set of real numbers. In addition, definitions are highlighted rather than being embedded in the prose of the text. Also, a new chapter () has been added to the text.

It gives a very brief discussion of Sylow Theorems and the Galois group. The text starts with the Peano postulates for the natural numbers in , with the various number systems of elementary algebra being constructed and their salient properties discussed. This not only introduces the reader to a detailed and rigorous development of these number systems but also provides the reader with much needed practice for the reasoning behind the properties of the abstract systems which follow. The first abstract algebraic system the Group is considered in ends with the JordanHlder Theorem for finite groups. Rings, Integral Domains Division Rings, Fields are discussed in . Throughout these chapters, considerable attention is given to finite rings.

Vector spaces are introduced in and other examples briefly considered. In the final chapter (), Boolean algebras are introduced and important applications to simple electric circuits are discussed. The co-author wishes to thank the staff of the Schaums Outlines group, especially Barbara Gilson, Maureen Walker, and Andrew Litell, for all their support. In addition, the co-author wishes to thank the estate of Dr. Frank Ayres, Jr. LLOYD R. LLOYD R.

JAISINGH

CONTENTS
Picture 3Picture 4SCHAUMS OUTLINE OF
Theory and Problems of
ABSTRACT ALGEBRA
CHAPTER 1
Sets
INTRODUCTION
In this chapter, we study the concept of sets. Specifically, we study the laws of operations with sets and Venn diagram representation of sets.
1.1 SETS
Any collection of objects as (a) the points of a given line segment, (b) the lines through a given point in ordinary space, (c) the natural numbers less than 10, (d) the five Jones boys and their dog, (eNext page
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