Lloyd R. Jaisingh Professor of Mathematics Ph.D. - Schaum’s Outline of Abstract Algebra

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

Second EditionFRANK AYRES, Jr., Ph.D
LLOYD R. JAISINGHProfessor of Mathematics
Morehead State University Schaums Outline Series 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. 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
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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

SCHAUMS OUTLINE OF
Theory and Problems ofABSTRACT ALGEBRA In this chapter, we study the concept of sets. Specifically, we study the laws of operations with sets and Venn diagram representation of 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