A group is a set of elements with a binary operation defined on the set.
(A binary operation on a set S is a function from S x S -> S.)

The binary operation must satisfy three additional properties:

1. There must be an _identity element_.

   e is an identity element if

   for all a, 

        e * a = a
   and  a * e = a

2. Every element must have an _inverse_

   b is an inverse of a if

        a * b = e
   and  b * a = e

3. The group operation must be _associative)

   For all a, b, and c, 

        (a * b) * c = a * (b * c)