9.1 Introduction to Group Theory

Group theory is a fundamental concept in abstract algebra, exploring sets with binary operations that satisfy specific properties. It introduces key ideas like closure, associativity, identity, and inverse elements, which form the foundation for understanding more complex algebraic structures.

This section focuses on different types of groups, including Abelian and cyclic groups, and examines important characteristics like order and subgroups. It also introduces group homomorphisms and isomorphisms, which help us compare and analyze different group structures.

Definition and Properties of Groups

Fundamental Group Concepts

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• Group consists of a set of elements and a binary operation that combines two elements to produce a third element within the set
• Binary operation maps two elements of a set to another element in the same set (function $f: G \times G \rightarrow G$)
• Closure property ensures the result of the binary operation on any two elements in the group remains within the group
• Associativity allows grouping of elements in different ways without changing the result ($a * (b * c) = (a * b) * c$)

Identity and Inverse Elements

• Identity element acts as a neutral element in the group, leaving other elements unchanged when combined with it ($e * a = a * e = a$)
• Inverse element, when combined with its corresponding element, produces the identity element ($a * a^{-1} = a^{-1} * a = e$)
• Every element in a group must have a unique inverse within the group
• Identity and inverse elements play crucial roles in maintaining the group structure

Special Types of Groups

Abelian Groups

• Abelian group exhibits commutativity, where the order of elements in the operation doesn't affect the result ($a * b = b * a$)
• Also known as commutative groups
• Examples include integers under addition ($\mathbb{Z}, +$) and real numbers under multiplication ($\mathbb{R}^*, \times$)
• Abelian groups have simpler structures and properties compared to non-Abelian groups
• Many theorems in group theory apply specifically to Abelian groups

Cyclic Groups

• Cyclic group can be generated by repeatedly applying the group operation to a single element, called the generator
• All elements in a cyclic group can be expressed as powers of the generator ($\{a^n : n \in \mathbb{Z}\}$)
• Can be finite or infinite depending on the order of the generator
• Finite cyclic groups include the group of rotational symmetries of a regular polygon
• Infinite cyclic groups include the integers under addition, with 1 or -1 as the generator

Group Characteristics

Order and Subgroups

• Order of a group refers to the number of elements in the group
• Finite groups have a countable number of elements, while infinite groups have uncountable elements
• Order of an element $a$ is the smallest positive integer $n$ such that $a^n = e$ (identity element)
• Lagrange's Theorem states that the order of a subgroup divides the order of the group
• Subgroups are subsets of a group that form a group under the same operation

Group Homomorphisms and Isomorphisms

• Group homomorphism is a function between two groups that preserves the group operation ($f(a * b) = f(a) * f(b)$)
• Isomorphism is a bijective homomorphism, indicating that two groups have the same structure
• Kernel of a homomorphism is the set of elements that map to the identity element in the codomain
• First Isomorphism Theorem relates quotient groups to homomorphic images