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.