We first show that every simple group of order 60 has to be isomorphic to A_5.
Next, for a given any prime p, we constructed two nonabelian groups, by means of semidirect product, of order p^3. We show that when p is odd then the above two groups are nonisomorphic, where as for p=2 both coincide with D_8.
Next we prove that, if p is an odd prime, then every nonabelian group of order p^3 must be isomorphic to one of the aforementioned two groups. Thus we classify all nonebelian groups of order p^3 for any odd prime p.
