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

- CLOSURE For all elements $a$ and $b$ in the group, the element $a*b$ is also in the group.
- ASSOCIATIVITY If $a, b$ and $c$ are in the group then $(a*b)*c = a*(b*c)$.
- IDENTITY The group contains an element $e$, called the identity, such that if $a$ is in the group then $a*e = e*a = a$.
- 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 group, $a*b=b*a$.