Group Theory

Chapter 2: Definitions and Examples

2.1 Introduction to Definitions and Examples
In this chapter, we will delve into the foundational concepts of group theory, starting with definitions and examples. Understanding the definitions of key terms and exploring various examples will provide you with a solid understanding of the structure and behavior of groups.

2.2 Definitions
2.2.1 Group
A group is a mathematical structure consisting of a set G and an operation (often denoted as *) that combines any two elements of G to produce another element in G. The operation must satisfy the following properties:
Closure: For any a, b ∈ G, the result of the operation a * b is also an element of G.
Associativity: For any a, b, c ∈ G, (a * b) * c = a * (b * c).
Identity Element: There exists an element e ∈ G, called the identity element, such that for any a ∈ G, a * e = e * a = a.
Inverse Element: For every element a ∈ G, there exists an element a⁻¹ ∈ G, called the inverse of a, such that a * a⁻¹ = a⁻¹ * a = e.

2.2.2 Subgroup
A subgroup of a group G is a subset H of G that forms a group itself with respect to the same operation as G. A subgroup must satisfy the following properties:
Closure: For any a, b ∈ H, the result of the operation a * b is also an element of H.
Associativity: For any a, b, c ∈ H, (a * b) * c = a * (b * c).
Identity Element: The identity element of G is also an element of H.
Inverse Element: For every element a ∈ H, its inverse a⁻¹ is also an element of H.

2.2.3 Abelian Group
An abelian group, also known as a commutative group, is a group in which the operation is commutative. That is, for any a, b ∈ G, a * b = b * a.

2.2.4 Cyclic Group
A cyclic group is a group that can be generated by a single element. In other words, there exists an element g ∈ G such that every element of G can be expressed as a power of g. The element g is called a generator of the cyclic group.

2.3 Examples
2.3.1 The Integers under Addition
The set of integers ℤ, together with the addition operation (+), forms a group. It satisfies all the properties of a group, including closure, associativity, identity (0), and inverse (-a for every a).

2.3.2 The Non-Zero Rational Numbers under Multiplication
The set of non-zero rational numbers ℚ0, with the multiplication operation (×), forms a group. It satisfies all the group properties, including closure, associativity, identity (1), and inverse (1/a for every non-zero rational number a).

2.3.3 Symmetric Group
The symmetric group, denoted by Sym(n), consists of all possible permutations of n elements. The group operation is the composition of permutations. The symmetric group is an important example in group theory and plays a significant role in understanding the structure of groups.

2.3.4 Klein Four-Group
The Klein four-group, denoted by V or V₄, is a group with four elements 'e, a, b, c'. The group operation is defined by the following table:
* e a b c
e e a b c
a a e c b
b b c e a
c c b a e
The Klein four-group is an example of an abelian group.

2.3.5 Dihedral Group
The dihedral group, denoted by Dₙ, is a group of symmetries of a regular n-gon. It consists of rotations and reflections. The group operation is composition. The dihedral group is an important example in understanding symmetry and group actions.

2.4 Conclusion
In this chapter, we explored the foundational definitions of group theory, including the definition of a group, subgroup, abelian group, and cyclic group. We also examined several examples, such as the integers under addition, the symmetric group, the Klein four-group, and the dihedral group. These examples provide a glimpse into the diverse structures and properties that can arise within groups. In the following chapters, we will further explore the properties, theorems, and applications of group theory.