Basic Abstract Algebra

For Graduate Students and Advanced Undergraduates

Robert B. Ash

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486318110
• Topic: Mathématiques
• Subtopic: Algèbre abstraite

Geared toward upper-level undergraduates and graduate students, this text surveys fundamental algebraic structures and maps between these structures. Its techniques are used in many areas of mathematics, with applications to physics, engineering, and computer science as well. Author Robert B. Ash, a Professor of Mathematics at the University of Illinois, focuses on intuitive thinking. He also conveys the intrinsic beauty of abstract algebra while keeping the proofs as brief and clear as possible.
The early chapters provide students with background by investigating the basic properties of groups, rings, fields, and modules. Later chapters examine the relations between groups and sets, the fundamental theorem of Galois theory, and the results and methods of abstract algebra in terms of algebraic number theory, algebraic geometry, noncommutative algebra, and homological algebra, including categories and functors. An extensive supplement to the text delves much further into homological algebra than most introductory texts, offering applications-oriented results. Solutions to all problems appear in the text.

Chapter 1

Group Fundamentals

1.1 Groups and Subgroups

1.1.1 Definition

A group is a nonempty set G on which there is defined a binary operation (a, b) → ab satisfying the following properties.

Closure: If a and b belong to G, then ab is also in G;

Associativity: a(bc) = (ab)c for all a, b, c ∈ G;

Identity: There is an element 1 ∈ G such that a1 = 1a = a for all a in G;

Inverse: If a is in G, then there is an element a⁻¹ in G such that aa⁻¹ = a⁻¹a = 1.

A group G is abelian if the binary operation is commutative, i.e., ab = ba for all a, b in G. In this case the binary operation is often written additively ((a, b) → a + b), with the identity written as 0 rather than 1.

There are some very familiar examples of abelian groups under addition, namely the integers

, the rationals

, the real numbers

, the complex numers

, and the integers

_m modulo m. Nonabelian groups will begin to appear in the next section.

The associative law generalizes to products of any finite number of elements, for example, (ab)(cde) = a(bcd)e. A formal proof can be given by induction. If two people A and B form a₁ … a_n in different ways, the last multiplication performed by A might look like (a₁ … a_i)(a_i+1 … a_n), and the last multiplication by B might be (a₁ … a_j)(a_j+1 … a_n). But if (without loss of generality) i < j, then (induction hypothesis)

and

.

By the n = 3 case, i.e., the associative law as stated in the definition of a group, the products computed by A and B are the same.

The identity is unique (1′ = 1′1 = 1), as is the inverse of any given element (if b and b′ are inverses of a, then b = 1b = (b′a)b = b′(ab) = b′1 = b′). Exactly the same argument shows that if b is a right inverse, and a a left inverse, of a, then b = b′.

1.1.2 Definitions and Comments

A subgroup H of a group G is a nonempty subset of G that forms a group under the binary operation of G. Equivalently, H is a nonempty subset of G such that if a and b belong to H, so does ab⁻¹. (Note that 1 = aa⁻¹ ∈ H; also, ab = a((b⁻¹)⁻¹) ∈ H.)

If A is any subset of a group G, the subgroup generated by A is the smallest subgroup containing A, often denoted by 〈A〉. Formally, 〈A〉 is the intersection of all subgroups containing A. More explicitly, 〈A〉 consists of all finite products a₁ … a_n, n = 1, 2, …, where for each i, either a_i or

belongs to A. To see this, note that all such products belong to any subgroup containing A, and the collection of all such products forms a subgroup. In checking that the inverse of an element of 〈A〉 also belongs to 〈A〉, we use the fact that

which is verified directly:

.

1.1.3 Definitions and Comments

The groups G₁ and G₂ are said to be isomorphic if there is a bijection f : G₁ → G₂ that preserves the group operation, in other words, f(ab) = f(a) f(b). Isomorphic groups are essentially the same; they differ only notationally. Here is a simple example. A group G is cyclic if G is generated by a single element: G = 〈a〉. A finite cyclic g...