A group is a simple mathematical system, so basic that groups appear wherever one looks in mathematics. Despite the primitive nature of a group, mathematicians have developed a rich theory about them. Specifically, a group is a mathematical system consisting of a set G and a binary operation * which has the following properties:

[1] x*y is in G whenever x and y are in G (closure).

[2] (x*y)*z = x*(y*z) for all x, y, and z in G (associative property).

[3] There exists and element, e, in G such that e*x=x*e=x for all x in G (existence of an identity element).

[4] For any element x in G, there exists an element y such that x*y=y*x=e (existence of inverses).

Note that commutativity is not required. That is, it need not be true that x*y=y*x for...