To Study on Number of Subgroups of Finite Abelian Group Zm ⊗ Zn

In this paper, we determine the number of subgroups of group Z_ (pᵐ) ⊗ Z_ (pⁿ) which may be cyclic or non –cyclic by using simple number –theoretic formulae. We describe the subgroups of the group and derive a simple formula for the total number s (m, n) of the subgroups where m, n are arbitrary positive integers. We point out that certain multiplicative properties of related counting functions for finite. Abelian groups are immediate consequences of these formula.