On the total number of principal series of a finite abelian group

TL;DR

This paper provides a bijective proof for counting the total number of principal series in finite abelian groups, simplifying the process and enabling easier generalization beyond specific cases.

Contribution

It introduces a new bijective proof method for counting principal series, improving upon previous direct calculation approaches for finite abelian groups.

Findings

Explicit formula for the number of principal series in certain finite abelian groups

A bijective proof method that simplifies counting

Enhanced generalizability to arbitrary finite abelian groups

Abstract

In this note we give a bijective proof for the explicit formula giving the total number of principal series of the direct product $\mathbb{Z}_{p^{\alpha_1}} \times \mathbb{Z}_{p^{\alpha_2}}$, where $p$ is a prime number. This new proof is easier to generalize to arbitrary finite abelian groups than the original direct calculation method.