is a set together with
a binary operation :
that is, a uniquely determined element
is assigned to each pair of elements where .
Furthermore the operation must satisfy the following requirements
- There exists a neutral element (or identity element):
It can be proved that the identity element is uniquely determined.
- There exists a reciprocal for each element of :
The element is called inverse element of . This inverse
element is uniquely determined and is often denoted by .
A group is called commutative or Abelian group, if its operation is
If it is clear which operation is used, then often only is written instead