Abstract Algebra
eBook - ePub

Abstract Algebra

Derek J.S. Robinson

Share book
  1. 347 pages
  2. English
  3. ePUB (mobile friendly)
  4. Available on iOS & Android
eBook - ePub

Abstract Algebra

Derek J.S. Robinson

Book details
Book preview
Table of contents

About This Book

This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and in the information and physical sciences. In addition to introducing the main concepts of modern algebra, the book contains numerous applications, which are intended to illustrate the concepts and to convince the reader of the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems and error correcting codes are described. Another feature of the book is that group theory and ring theory are carried further than is often done at this level. There is ample material here for a two semester course in abstract algebra.

The importance of proof is stressed and rigorous proofs of almost all results are given. But care has been taken to lead the reader through the proofs by gentle stages. There are nearly 400 problems, of varying degrees of difficulty, to test the reader's skill and progress. The book should be suitable for students in the third or fourth year of study at a North American university or in the second or third year at a university in Europe, and should ease the transition to (post)graduate studies.

Frequently asked questions

How do I cancel my subscription?
Simply head over to the account section in settings and click on “Cancel Subscription” - it’s as simple as that. After you cancel, your membership will stay active for the remainder of the time you’ve paid for. Learn more here.
Can/how do I download books?
At the moment all of our mobile-responsive ePub books are available to download via the app. Most of our PDFs are also available to download and we're working on making the final remaining ones downloadable now. Learn more here.
What is the difference between the pricing plans?
Both plans give you full access to the library and all of Perlego’s features. The only differences are the price and subscription period: With the annual plan you’ll save around 30% compared to 12 months on the monthly plan.
What is Perlego?
We are an online textbook subscription service, where you can get access to an entire online library for less than the price of a single book per month. With over 1 million books across 1000+ topics, we’ve got you covered! Learn more here.
Do you support text-to-speech?
Look out for the read-aloud symbol on your next book to see if you can listen to it. The read-aloud tool reads text aloud for you, highlighting the text as it is being read. You can pause it, speed it up and slow it down. Learn more here.
Is Abstract Algebra an online PDF/ePUB?
Yes, you can access Abstract Algebra by Derek J.S. Robinson in PDF and/or ePUB format, as well as other popular books in Mathematics & Algebra. We have over one million books available in our catalogue for you to explore.


De Gruyter

1 Sets, relations and functions

The concepts introduced in this chapter are truly fundamental and underlie almost every branch of mathematics. Most of the material is quite elementary and will be familiar to many readers. Nevertheless readers are encouraged to review the material and to check notation and definitions. Because of its nature the pace of this chapter is somewhat faster than in subsequent chapters.

1.1 Sets and subsets

By a set we shall mean any well-defined collection of objects, which are called the elements of the set. Some care must be exercised in using the term “set” because of Bertrand Russell’s famous paradox, which shows that not every collection can be regarded as a set. Russell considered the collection C of all sets which are not elements of themselves. If C is allowed to be a set, a contradiction arises when one inquires whether or not C is an element of itself. Now plainly there is something suspicious about the idea of a set being an element of itself and we shall take this as evidence that the qualification “well-defined” needs to be taken seriously. A collection that is not a set is called a proper class.
Sets will be denoted by capital letters and their elements by lower case letters. The standard notation
a ∈ A
means that a is a element of the set A, or a belongs to A. The negation of aA is denoted by a ∉ A. Sets can be defined either by writing their elements out between braces, as in {a, b, c, d}, or alternatively by giving a formal description of the elements, the general format being
A = {a | a has property P},
i.e., A is the set of all objects with the property P. If A is a finite set, the number of its elements is written
Subsets. Let A and B be sets. If every element of A is an element of B, we write
A ⊆ B
and say that A is a subset of B, or that A is contained in B. If AB and BA, so that A and B have exactly the same elements, then A and B are said to be equal,
A = B.
The negation of this is A ≠ B. The notation AB is used if AB and A ≠ B; then A is called a proper subset of B.

Some special sets. A set with no elements at all is called an empty set. An empty set E is a subset of any set A; for if this were false, there would be an element of E that is not in A, which is certainly wrong. As a consequence, there is exactly one empty set: for if E and E′ are two empty sets, then EE′ and E’E, so that E = E′. The unique empty set is written
Some fur...

Table of contents