1. Introduction

Group

A collection of "things" (numbers, functions, vectors, symmetries, etc)

For a set to be considered a Group, it must satisfy all of the composition rules.

 

Composition Rules

A group $G$ is a set of elements $\{x, y, z, ...\}$ together with an operation $\cdot$ that satisfies the following properties

• Closure - For any $x, y \in G,$ the product $(x \cdot y) \in G$
• Associativity - For all elements in $G$, $(x \cdot y) \cdot z = x \cdot (y \cdot z)$
• Identity - There exists an element $e \in G$ such that $e \cdot x = x \cdot e = x$ for every $x \in G$
• Inverse - For each $x \in G$, there exists an element $x^{-1} \in G$ such that $x^{-1} \cdot x = x \cdot x^{-1} = e$

we will be written $e$ as $Id$ from now on

 

Abelian Group

If two elements $x, y \in G$ are cummutative such that

$$x \cdot y = y \cdot x$$

then the group is said to be abelian

Examples

• The groups $\mathbb{Z}, \; \mathbb{Q}, \; \mathbb{R}, \; \mathbb{C}$ with addition are cummutative

• The groups $\mathbb{R}^*, \; \mathbb{C}^*$ are commutitive with multiplication

The $(^*)$ indicates all non-zero elements, since $0$ doesn't not have a multiplicative inverse so it would not satisfy the Inverse composition rule