A group in mathematics has the following properties:

There exist a set of elements p, q, r, …  and a binary operation which applied to p, q gives pq.

The set is closed under this single-valued operation.

1.  The associative law: p(qr) = (pq)r.
2. Identity law: p i = i p.
3. Inverse law:   there exist p’ such that p p’ = p’ p 

If the group axioms are full-filled i is unique and the inverse is also unique.