# group

## Table of Contents

## 1. Definition

A group is an ordered pair \((G, *)\) where \(G\) is a set and \(*\) is a binary operation (operation defined between two members of set G) defined such that:

\begin{align*} a * b \in G \\ \exists e : a * e = a \end{align*}
where the operation \(*\) is said to be closed under \(G\), and \(e\) is called the *identity* of group \((G, *)\).

### 1.1. Associativity

This is the property such that:

\begin{align*} (a * b) * c = a * (b * c) \end{align*}### 1.2. inverse

An inverse is defined as follows:

\begin{align*} \forall a \exists a^{-1} : a * a^{-1} = e \end{align*}## 2. Motivation

In physics, natural phenomena including conservation laws follow from group symmetries.