Group

A group is a set of elements $\mathcal{G}$ along with a binary operation $\star$ which satisfy the following proterties:

Closure: For all $a,b\in\mathcal{G}$ , $a\star b\in\mathcal{G}$ .
Associativity: For all $a,b,c\in\mathcal{G}$ , $a\star(b\star c)=(a\star b)\star c$ .
Identity: There is some element $e\in\mathcal{G}$ , called the identity element, such that for all $a\in\mathcal{G}$ , $e\star a=a=a\star e$ .
Inverse: For all $a\in\mathcal{G}$ , there is a corresponding $b \in \mathcal{G}$ , called the inverse of $a$ , such that $a\star b=e=b\star a$ . We often write $b$ as " $- a$ " or " $a^{-1}\,$ ".