# Alternate Definition of a Group

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 .

Thus

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.