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group

Table of Contents

1. Definition

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

a*b∈G∃e:a*e=a\begin{aligned}a * b \in G \\\exists e : a * e = a\end{aligned}

where the operation ** is said to be closed under GG, and ee is called the identity of group (G,*)(G, *).

1.1. Associativity

This is the property such that:

(a*b)*c=a*(b*c)\begin{aligned}(a * b) * c = a * (b * c)\end{aligned}

1.2. inverse

An inverse is defined as follows:

∀a∃a−1:a*a−1=e\begin{aligned}\forall a \exists a^{-1} : a * a^{-1} = e\end{aligned}

2. Motivation

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