Abstract Algebra

Derek J.S. Robinson

• Publisher: De Gruyter
• Year: 2015
• ISBN:
  9783110385601
• Edition: 2
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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.

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 a ∈ A 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

|A|.

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 A ⊆ B and B ⊆ A, 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 A ⊂ B is used if A ⊆ B 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 E ⊆ E′ and E’ ⊆ E, so that E = E′. The unique empty set is written

Some fur...