Counting subgroups of fixed order in finite abelian groups

Abstract

We use recurrence relations to derive explicit formulas for counting the number of subgroups of given order (or index) in rank 3 finite abelian p-groups and use these to derive similar formulas in few cases for rank 4. As a consequence, we answer some questions by M. Tärnäuceanu in [24] and L. Tóth in [25]. We also use other methods such as the method of fundamental group lattices introduced in [24] to derive a similar counting function in a special case of arbitrary rank finite abelian p-groups.