Let G be a finite nonempty set with an operation * such that:

1. G is closed under *.

2. * is associative.

3. Given a,b,c in G with a*b=a*c, then b=c.

4. Given a,b,c in G with b*a=c*a, then b=c.

Prove that G must be a group under *.

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It's obvious that identity element satisfies the conditions 3 and 4, but I don't know whether that proves that the identity element is contained in G or not? moreover, How can I show that the inverse of any element in G is contained in G?