A group is a set of elements together with a binary operation (which we denote here by *) with the following properties:

(1) CLOSURE For all elements $a$ and $b$ in the group, the element a*b is also in the group.

(2) ASSOCIATIVITY If $a, b$ and $c$ are in the group then $(a * b) * c = a * (b * c)$ .

(3) IDENTITY The group contains an element $e$ , called the identity, such that if $a$ is in the group then $a * e = e * a = a$ .

(4) INVERSES If $a$ is an element in the group then there is also an element in the group $a'$ , called the inverse of $a$ , such that $a * a' = a' * a = e$ .

Some groups, which are called COMMUTATIVE or ABELIAN, have the property that, for all pairs of elements in the set, $a * b = b * a$ .