Definition of a Group

A group is a set $G$ together with an operation $\ast$ such that all of four of the following axioms hold:

1. Closure: If $x,y\in G$, then $x\ast y$ is also in $G$.
2. Associativity: If $x,y\in G$, then $x\ast (y\ast z)=(x\ast y)\ast z$.
3. Existence of an Identity Element: We can find an element $e\in G$ satisfying $e\ast x=x\ast e=x$, for all $x\in G$ .
4. Existence of inverse elements: If $x\in G$, we can find $y\in G$ such that $x\ast y=y\ast x=e$.