A set with binary composition denoted multiplicatively is a group if
(i) The composition is associative.
(ii) For every pair of elements , the equations and have unique solutions in .
Proof: Binary operation implies that the set under consideration is closed under the operation. Now to prove that the set is a group, we have to show the left identity exists and each element of possesses a left inverse with respect to the operation under consideration.
It is given that for every pair of elements the equation has a solution in . Therefore, say such that .
Now, let us suppose that is any arbitrary element of . Therefore, there exists such that .
Therefore there exists such that
is the left identity.
Now, let be an element. is the inverse of in .
Let such that , then as has a solution in .
Thus is the left inverse of in . Therefore each element of possesses a left inverse. Hence is a group for the given composition if the postulate and (ii) are satisfied.