1. The problem statement, all variables and given/known data 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 *. 3. The attempt at a solution 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?