Integral of groups

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 Sn for n ≥ 3, generalized quaternion group Q2n for n ≥ 4 and Dihedral group Dn for n ≥ 3 cannot occur as commutator subgroup of any group. Moreover, we characterize all finite groups with the cyclic commutator subgroup isomorphic to ℤ2n or ℤ3n.