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.