Abelian group

An  is an algebraic structure. A abelian group consists of an underlying set of objects, $$G$$, and a binary operation, $$*$$, and satisfies the following properties:


 * 1) Closure: For all $$x,y \in G$$ we have $$x*y\in G$$
 * 2) Associativity: For all $$x,y,z \in  $$ we have that $$x*(y*z)=(x*y)*z$$.
 * 3) Commutativity: For all $$x,y\in G$$ we have that $$x*y=y*x$$
 * 4) Identity: There is an element $$e\in G$$ such that for all $$ g\in G$$ we have that $$e*x=x*e=x$$
 * 5) Inverse: For all $$g\in G$$ there is an element $$g^{-1}\in G$$ such that $$g*g^{-1}=g^{-1}*g=e$$

The third property, commutativity, distinguishes an abelian group from a non-abelian group.