In this paper, we define the concept of integral for an arbitrary group, and we try to answer this question: Which groups can be considered as a commutator subgroup? For instance, we will show that symmetric group S-n for n >= 3, generalized quaternion group Q(2)n for n >= 4 and Dihedral group D-n for n >= 3 cannot occur as commutator subgroup of any group. Moreover, we characterize all finite groups with the cyclic commutator subgroup isomorphic to Z(2)n Or Z(3)n.