Counting subgroups of fixed order in finite abelian groups

TL;DR

This paper develops recurrence relations and formulas to count subgroups of fixed order in finite abelian p-groups of ranks 3 and 4, answering existing open questions and extending methods to arbitrary ranks.

Contribution

It introduces explicit formulas for subgroup counts in rank 3 and 4 finite abelian p-groups and applies the fundamental group lattice method to broader cases.

Findings

Derived explicit formulas for rank 3 subgroup counts.

Extended formulas to some rank 4 cases.

Answered open questions by Trnceanu and Tth.

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. T$\ddot{a}$rn$\ddot{a}$uceanu in \cite{MT} and L. T$\dot{\acute{o}}$th in \cite{LT}. We also use other methods such as the method of fundamental group lattices introduced in \cite{MT} to derive a similar counting function in a special case of arbitrary rank finite abelian p-groups.