Introduction to Abstract Algebra

When we study mathematics, we study various structures, such as numbers, vectors, and functions. In elementary mathematics, we learn about functions of real numbers, such as polynomials, trigonometric functions, exponentials, logarithms, and so on. In linear algebra, we learn about vector spaces, which are sets of vectors that can be added together and multiplied by scalars. In calculus, we learn about limits, derivatives, and integrals of functions. In differential equations, we learn about equations that involve derivatives of functions. In probability theory, we learn about random variables and their distributions.

Is there a way to study these structures in a more general way? This is where abstract algebra comes in. Abstract algebra is the study of algebraic structures, such as groups, rings, and fields. These structures are defined by a set of axioms, which are rules that the elements of the structure must satisfy. By studying these structures in a more general way, we can gain a deeper understanding of the underlying principles that govern them.

Studying abstract algebra is an absolute must for quantum field theory, as it provides the mathematical framework for understanding symmetries, conservation laws, and the behavior of particles and fields at a fundamental level. In this section, we will introduce the basic concepts of abstract algebra, including groups, rings, and fields.

Table of Contents

• Table of Contents
• Fundamental Ideas
  • Equivalence Relations
  • Partitions
• Groups
  • Example: Number Line
  • Example: Rotation of a Square
  • Group Homomorphisms
  • Group Actions
  • Subgroups and Quotient Groups
• Rings and Fields

Fundamental Ideas

At its core, abstract algebra is the study of mathematical structures and the relationships between them. The most fundamental concepts relate to set theory. A set is a collection of distinct objects, which can be anything: numbers, letters, shapes, or even other sets. It must be well-defined, meaning that it is clear whether an object belongs to the set or not. We can perform various operations on sets, such as union, intersection, and difference.

Given two sets and , their Cartesian product is the set of all ordered pairs where and . We often use Cartesian products in defining more complex structures. For example, you have probably seen functions denoted as , which means that the function takes in two inputs, one from set and one from set , and produces an output in set . In this notation, the domain of the function is the set of all possible ordered pairs where and .

A subset of is known as a relation from to . For example, the "less than" relation on the set of real numbers is a subset of , consisting of all ordered pairs such that . A function is a special type of relation where each element in the domain is associated with exactly one element in the codomain. For instance, the function is a relation, a subset of , where each real number is paired with its square . Instead of denoting a function as a subset of , we often use the notation to indicate that is a function from set to set .

Equivalence Relations

In mathematics, we often want to group objects that are "similar" in some way. In elementary algebra, we have the equality sign , which indicates that two objects are exactly the same. Is there a way to generalize this idea of equality to a broader concept of similarity? The answer to this question lies in the concept of an equivalence relation.

An equivalence relation is a relation that satisfies three properties (these should make sense intuitively):

1. Reflexivity: For any element in the set, is related to itself. i.e., .
2. Symmetry: If is related to , then is related to . i.e., if , then .
3. Transitivity: If is related to and is related to , then is related to . i.e., if and , then .

Written in the language of set theory, an equivalence relation on a set is a relation such that for all ,

1. Reflexivity: .
2. Symmetry: If , then .
3. Transitivity: If and , then .

Another important concept related to equivalence relations is that of an equivalence class. Given an element in the set , the equivalence class of is the set of all elements in that are related to . It is denoted by and is defined as

Partitions

Suppose we have a set and an equivalence relation on . Then, let's split the set into subsets that don't overlap (i.e., they are disjoint) and cover the entire set . This collection of subsets is called a partition of the set .

Formally speaking, a partition of a set is a collection of non-empty subsets of such that

One common application of a partition is in defining the Riemann integral. Recall from calculus that we can approximate the area under a curve by dividing the interval into smaller subintervals and summing up the areas of rectangles. These intervals are precisely a partition of the interval.

There is a deep connection between equivalence relations and partitions. Given an equivalence relation on a set , we can form a partition of by taking the equivalence classes of each element in . Conversely, given a partition of a set , we can define an equivalence relation on by saying that two elements and are related if they belong to the same subset in the partition. This means that there is a one-to-one correspondence between equivalence relations and partitions.

Groups

A group is a set of elements that can be combined using a binary operation, such as addition or multiplication.

Consider an equilateral triangle in the plane. There are a few ways to transform the triangle, such as rotating it or reflecting it. Group theory is the study of such transformations.

In particular, we will study the set of all transformations that can be applied to the triangle, and how these transformations can be combined. Groups fundamentally represent different ways to transform an object. As such, we naturally expect certain properties to hold;

1. There is always one transformation that does nothing to the object. For the rotation group for the triangle, for example, it is akin to rotating the triangle by degrees. This is called the identity transformation.
2. The transformations can be combined to form new transformations. For example, if we rotate the triangle by degrees and then by another degrees, we can combine these two transformations to form a new transformation that rotates the triangle by degrees. This is called the closure property.
3. The way we combine the transformations is associative. For example, if we rotate the triangle by degrees and then reflect it, we can combine these two transformations in any way to get the same result. This is called the associative property.
4. Each transformation has an inverse transformation that undoes it. For example, if we rotate the triangle by degrees, we can undo this transformation by rotating it by degrees. This is called the inverse property.

Given that we expect these properties to hold, we can define a group as follows:

Group: A group is a set together with a binary operation that combines two elements to form a third element, such that the following properties hold:

1. Closure: For all , .
2. Associativity: For all , .
3. Identity: There exists an element such that for all , .
4. Inverse: For each , there exists an element such that .

In this definition, is the set of elements, and is the binary operation that combines the elements. and are elements of the group, which in our example are the rotations of the triangle. is the identity element, which in our example is the rotation by degrees.

There are some properties of groups that are worth noting, many of which should be easy to verify from the definition:

• The identity element is unique.

  Proof: Suppose there are two identity elements and in the group. Then, for any element , we have and . Therefore, for all . (By transitivity of equality) In particular, if we take , we have . But since is an identity element, we also have . Therefore, , which means that the identity element is unique. (By transitivity of equality)

• The inverse element of each element is unique.

  Proof: Suppose there are two inverse elements and for an element . Then, we have and . Therefore, . By the cancellation property of groups, we can cancel from both sides to get , which means that the inverse element is unique.

• The inverse of the identity element is the identity element itself.

  Proof: Let be the identity element of the group. Then, we have . Therefore, is its own inverse.

Example: Number Line

Consider the set of all integers . When two numbers are added, this can be visualized as moving the number line twice. For example, if we are at and we add , we shift the number line to the left by ;

When we add multiple numbers, this is just a matter of moving the number line multiple times, which can be combined into a single movement. For example, if we add and then , we can combine these two movements into a single movement of .

When we visualize addition like this, this gives us a clue that addition is an operation that satisfies the properties of a group. While it can be a good exercise to algebraically verify that addition satisfies the properties of a group, it is more insightful to notice the link between addition and its geometric representation. In particular, we can see that the number line is a one-dimensional space, and the operation of addition is a way to move along this space. This allows us to extend this group to higher dimensions, eventually leading to the concept of a vector space.

In addition to the properties of a group, addition is also commutative. In other words, the order in which we add the numbers does not matter. Groups that are commutative are called abelian groups.

Example: Rotation of a Square

Consider the set of all rotations of a square. We can rotate the square by , , , and degrees.

We can see that the set of all rotations of a square satisfies the properties of a group.

1. Closure: If we rotate the square by degrees and then by degrees, we can combine these two transformations to form a new transformation that rotates the square by degrees.
2. Associativity: The way we combine the transformations is associative. For example, if we rotate the square by degrees and then reflect it, we can combine these two transformations in any way to get the same result.
3. Identity: The identity transformation is the rotation by degrees.
4. Inverse: The inverse transformation of a rotation by degrees is a rotation by degrees.

Additionally, the set of all rotations of a square is also abelian, since the order in which we apply the transformations does not matter.

Group Homomorphisms

Suppose we have two groups and . Is there a way to relate these two groups? If we can find a way to relate the two groups, we can use the properties of one group to study the other group. We introduce something known as a homomorphism:

This is a mapping between the two groups that preserves the group structure:

where and is the group operation (for the group on the left and for the group on the right). This means that if we take two elements and from group , and we apply the group operation to them, we can map the result to an element in group .

More specifically, we can look for a one-to-one mapping between the two groups, known as a bijection. This means that we can map each element of to a unique element of , and vice versa. These bijective homomorphisms are called isomorphisms.

Group Actions

A group action is a way for a group to "act" on a set. Think of a group as a set of transformations, and the set as a set of objects that can be transformed. For example, consider the group of rotations of a square. This group can act on the set of vertices of the square.

To formalize this, we define a group action as follows:

Group Action: A left group action of a group on a set is a function such that for all and all :

1. Identity: , where is the identity element of .
2. Compatibility: .
3. Inverses: for all and .
4. Identity: , where is the identity element of .

Likewise, a right group action of a group on a set is a function such that for all and all :

1. Identity: , where is the identity element of .
2. Compatibility: .
3. Inverses: for all and .
4. Identity: , where is the identity element of .

In this definition, is the group, is the set, and is the group action. Let's consider an example.

The group of rotations in two dimensions, , can act on the set of points in the plane . The action is given by rotating the point around the origin by the angle .

This is a left group action, since the group element is on the left side of the point.

Subgroups and Quotient Groups

A subgroup is a subset of a group that is itself a group. For example, consider the group of all integers under addition. The set of even integers is a subgroup of , since it satisfies the group properties.

A coset is a subset of a group that is formed by multiplying all elements of a subgroup by a fixed element of the group. For example, consider the group of all integers under addition, and the subgroup of even integers . The coset of with respect to the element is the set of all odd integers .

A subgroup of a group is called a normal subgroup if it is invariant under conjugation by elements of . In other words, for all and ,

Rings and Fields

A ring is a set of elements equipped with two binary operations, typically called addition and multiplication, that satisfy certain properties. Rings generalize the concept of familiar number systems, such as integers and polynomials, by allowing for more abstract structures. A field is a special type of ring in which every non-zero element has a multiplicative inverse. In other words, a field is a ring in which division is always possible (except by zero).

Ring: A ring is a set equipped with two binary operations, addition () and multiplication (), such that the following properties hold:

1. Addition: is an abelian group.
2. Multiplication: Multiplication is associative; for all , .
3. Distributive Laws: For all , and .

Field: A field is a set equipped with two binary operations, addition () and multiplication (), such that the following properties hold:

1. Addition: is an abelian group.
2. Multiplication: is an abelian group, where is the additive identity.
3. Distributive Laws: For all , and .

Rings and fields are fundamental structures in abstract algebra, and they have numerous applications in various branches of mathematics, including number theory, algebraic geometry, and cryptography.