Groups

Definition

A group is a non-empty set $G$ together with a binary operation $*$ such that the following four conditions hold:

1. Closure

: For all $a, b \in G$ , the result $a * b$ is also in $G$ .

2. Associativity

: For all $a, b, c \in G$ , $(a * b) * c = a * (b * c)$

3. Identity element

: There exists an element $e \in G$ such that for every $a \in G$ , $e * a = a * e = a$

4. Inverse element

: For every $a \in G$ , there exists an element $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$

If the operation is also commutative, meaning $a * b = b * a$ for all $a, b \in G$ , then the group is called an abelian group.

Main Content

1. Group Axioms and Basic Structure

A group must satisfy exactly four core rules: closure, associativity, identity, and inverses.
These rules guarantee that the set behaves consistently under the operation, allowing calculations and transformations to be carried out without leaving the set.

Closure

Closure means that combining any two elements of the group always gives another element in the same group.
For example, the set of integers $\mathbb{Z}$ under addition is closed because the sum of any two integers is again an integer.

Example:

$3 + 5 = 8 \in \mathbb{Z}$
$(-2) + 7 = 5 \in \mathbb{Z}$

Associativity

Associativity means that when three elements are combined, the way in which they are grouped does not affect the result.
This is essential because it allows us to write expressions without worrying about parentheses.

Example:

$(2 + 3) + 4 = 2 + (3 + 4) = 9$

Not every operation is associative. For instance, subtraction is not associative:

$(5 - 3) - 1 = 1$
$5 - (3 - 1) = 3$

Identity and Inverse

The identity element acts as a neutral element, leaving every other element unchanged.
For addition, the identity is $0$ .
For multiplication, the identity is $1$ .

An inverse element is one that combines with a given element to produce the identity.
For addition, the inverse of $a$ is $-a$ .
For multiplication, the inverse of a nonzero real number $a$ is $\frac{1}{a}$ .

2. Types of Groups

Groups are classified based on whether the operation is commutative and on the nature of the set.
Common types include abelian groups, finite groups, infinite groups, cyclic groups, and permutation groups.

Abelian Groups

A group is abelian if the operation is commutative: $a * b = b * a$ for all $a, b \in G$ .

Example:

$(\mathbb{Z}, +)$ is abelian because $a+b=b+a$ .

Non-example:

The set of $2 \times 2$ invertible matrices under multiplication is generally not abelian.

Finite and Infinite Groups

A finite group has a limited number of elements.
An infinite group has infinitely many elements.

Examples:

Finite: symmetries of a triangle
Infinite: integers under addition

Cyclic Groups

A group is cyclic if all of its elements can be generated by repeatedly applying the operation to one element, called a generator.

Example:

$(\mathbb{Z}, +)$ is cyclic because every integer can be generated from $1$ or $-1$ : $\ldots, -2, -1, 0, 1, 2, \ldots$

Permutation Groups

These are groups formed by rearrangements of objects.
A permutation group is important in symmetry and combinatorics.

Example:

The different ways to arrange three objects form a group under composition.

3. Subgroups and Examples of Groups

A subgroup is a smaller group contained inside a larger group and using the same operation.
Studying subgroups helps break complicated groups into simpler parts and understand their internal structure.

Subgroup Idea

If $H$ is a subset of a group $G$ , then $H$ is a subgroup if it is itself a group under the same operation.

A useful subgroup test:

$H$ is non-empty.
For any $a, b \in H$ , the element $ab^{-1}\in H$ .

Common Examples of Groups

1. Integers under addition

: $(\mathbb{Z}, +)$
Closure: sum of integers is an integer
Identity: $0$
Inverse: $-a$

2. Nonzero real numbers under multiplication

: $(\mathbb{R}\setminus\{0\}, \times)$
Closure: product of nonzero reals is nonzero
Identity: $1$
Inverse: $\frac{1}{a}$

3. Symmetries of a square

Includes rotations and reflections
Forms a group under composition

4. Matrices

The set of all invertible $n \times n$ matrices forms a group under multiplication

Simple Structure of Symmetries of a Square

A square has 8 symmetries: 4 rotations and 4 reflections.

     r0
  ---------- 
 |          |
 |          |
 |          |
  ---------- 
 r90, r180, r270 and reflections across axes/diagonals

This group is often used to study geometric symmetry.

Working / Process

1. Identify the set and operation

Decide what objects are being studied and what operation combines them.
Example: integers with addition, nonzero real numbers with multiplication, transformations with composition.

2. Verify the group axioms

Check closure, associativity, identity, and inverses carefully.
If even one axiom fails, the structure is not a group.

3. Classify and analyze the group

Determine whether it is abelian, cyclic, finite, infinite, or has subgroups.
Use examples and properties to understand its structure and behavior.

Advantages / Applications

Groups help describe symmetry in geometry, nature, and art.
They are fundamental in solving algebraic equations and understanding polynomial roots.
They are widely used in physics, chemistry, and computer science, especially in particle theory, crystallography, coding theory, and cryptography.

Summary

Groups are algebraic structures with a set and one operation satisfying closure, associativity, identity, and inverses.
They provide a general framework for studying arithmetic, symmetry, and transformations.
Many important mathematical objects, such as integers under addition and symmetries of shapes, are groups.
Groups are foundational in advanced mathematics and its real-world applications.