Applied Abstract Algebra with MapleTM and MATLAB®

Richard Klima, Neil Sigmon, Ernest Stitzinger

• Publisher: Chapman and Hall/CRC
• Year: 2015
• Print ISBN: 9781032292083
• eBook ISBN: 9781482248289
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

Applied Abstract Algebra with MapleTM and MATLAB provides an in-depth introduction to real-world abstract algebraic problems. This popular textbook covers a variety of topics including block designs, coding theory, cryptography, and counting techniques, including Polya's and Burnside's theorems. The book also includes a concise review of all prereq

Chapter 1

Preliminary Mathematics

There are two purposes to this chapter. We very quickly and concisely review some of the basic algebraic concepts that are probably familiar to many readers, and also introduce some topics for specific use in later chapters. We will generally not pursue topics any further than necessary to obtain the material needed for the applications that follow. Topics reviewed in this chapter include permutation groups, the ring of integers, polynomial rings, finite fields, and examples that incorporate these topics using the philosophies of concepts covered in later chapters.

1.1Permutation Groups

Suppose a set G is closed under an operation *. That is, suppose a * b ∈ G for all a, b ∈ G. Then * is called a binary operation on G. We will use the notation (G, *) to represent the set G with this operation. Suppose (G, *) also satisfies the following three properties.

1. (a * b) * c = a * (b * c) for all a, b, c ∈ G.
2. There exists an identity element e ∈ G for which e * a = a * e = a for all a ∈ G. 3.
3. For each a ∈ G, there exists an inverse element b ∈ G for which a * b = b * a = e. The inverse of a is usually denoted by a⁻¹ if * is a general operation or multiplication, and −a if * is addition.

Then (G, *) is called a group. For example, it can easily be verified that for the set ℤ of integers, (ℤ, +) is a group with identity element 0, but (ℤ, ·) with normal integer multiplication is not a group.

Let S be a set, and let B(S) be the collection of all bijections (i.e., one-to-one and onto mappings) on S. Then any α ∈ B(S) can be uniquely expressed by its action α(s) on the elements s ∈ S.

Example 1.1 If A = {1, 2, 3}, then B(A) contains six elements. One α ∈ B(A) can be expressed as α(1) = 2, α(2) = 3, and α(3) = 1. □

Let ∘ represent the composition operation on B(S). Specifically, if α, β ∈ B(S), then define α ∘ β by the action (α ∘ β)(s) = α(β(s)) for s ∈ S. Since the composition of two bijections on S is also a bijection on S, then α ∘ β ∈ B(S). Thus, ∘ is a binary operation on B(S). It can easily be verified that (B(S), ∘) is a group.

A group (G, *) is said to be abelian or commutative if a * b = b * a for all a, b ∈ G. For example, since m + n = n + m for all m, n ∈ ℤ, then the group (ℤ, +) is abelian. However, for a set S with more than two elements, there do exist α, β ∈ B(S) such that α ∘ β ≠ β ∘ α. Thus, for a set S with more than two elements, the group (B(S), ∘) is not abelian....

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Preliminary Mathematics
• 2 Block Designs
• 3 Error-Correcting Codes
• 4 BCH Codes
• 5 Reed-Solomon Codes
• 6 Algebraic Cryptography
• 7 Vigenère Ciphers
• 8 RSA Ciphers
• 9 Elliptic Curve Cryptography
• 10 The Advanced Encryption Standard
• 11 Pólya Theory
• 12 Graph Theory
• 13 Symmetry in Western Music
• Bibliography
• Hints or Answers for Selected Exercises
• Index