Vlastimil Dlab and Kenneth S Williams - Invitation to Algebra: A Resource Compendium for Teachers, Advanced Undergraduate Students and Graduate Students in Mathematics

A Resource Compendium for Teachers, Advanced Undergraduate Students and Graduate Students in Mathematics

A Resource Compendium for Teachers, Advanced Undergraduate Students and Graduate Students in Mathematics

Vlastimil Dlab Kenneth S Williams

CarletonUniversity, Canada

Published by

World Scientific Publishing Co. Pte. Ltd.

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Library of Congress Control Number: 2020937550

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

INVITATION TO ALGEBRA

A Resource Compendium for Graduate Students and Advanced Undergraduate Students in Mathematics

Copyright 2020 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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ISBN 978-981-121-997-9 (hardcover)

ISBN 978-981-121-998-6 (ebook for institutions)

ISBN 978-981-121-999-3 (ebook for individuals)

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Desk Editor: Soh Jing Wen

Printed in Singapore

To our wives Helena and Carole

Up until towards the end of the nineteenth century, algebra loosely comprised the study of a variety of concrete systems such as vectors, polynomials, quaternions, and matrices. However, in the decade or so before the end of that century, mathematicians recognized that by abstracting the common content of such systems these different realms of algebra could be brought efficiently together. The foremost abstraction was to treat them as sets of elements subject to an operation specified by certain abstract properties. Thus, for example, the set of equivalence classes of binary quadratic forms ax2 + bxy+ cz2 treated by Gauss and the class of transformations

in the complex plane studied by Mbius were brought efficiently together under that part of abstract algebra known as the theory of groups.

Successful students know that learning an abstract subject like mathematics requires an appreciation of concrete examples which illustrate and motivate the underlying concepts. The learning process should imitate the way in which children learn new notions. A child does not learn what a table is through an abstract description of a certain construction but rather, after being shown many different tables, learns to single out tables from collections of furniture. In the same way an abstract notion should be built on well-chosen concrete examples. If a student is provided with a variety of concrete groups, then the definition of a group will arise naturally. The pedagogical principle which we shall follow is well expressed in a Chinese proverb attributed to Confucius (551 - 479 BCE)

I hear and I forget.