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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.
Khan et al. (Sun,) studied this question.