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# Group

A group is a pair $(G,*)$, where $G$ is a non-empty set and $*$ is a binary operation on $G$, such that the following conditions hold:

1. For any $a,b$ in $G$, $a*b$, belongs to $G$. ($G$ is closed under $*$).
2. For any $a,b,c\in G$, $(a*b)*c=a*(b*c)$. ($*$ is associative).
3. There is an element $e\in G$ such that $g*e=e*g=g$, for any $g\in G$. ($e$ is the identity element).
4. For any $g\in G$, there exists an element $h$ such that $g*h=h*g=e$. ($h=g^{-1}$ is the inverse of $g$).

Usually, the symbol $*$ is omitted and we write $ab$ for $a*b$. Sometimes, the symbol $+$ is used to represent the operation, when the group is abelian and, especially, in linear algebra.

Properties.

• there is only one identity element,
• for every element there is only one inverse.