An elementary proof of a power series identity for the weighted sum of all finite abelian p-groups

TL;DR

This paper provides a new elementary combinatorial proof for a known power series identity involving the weighted sum of finite abelian p-groups and their automorphisms, simplifying previous complex proofs.

Contribution

The paper introduces a straightforward combinatorial approach to prove a power series identity for finite abelian p-groups, previously established by Cohen and Lenstra.

Findings

Confirmed the power series identity using elementary combinatorial methods

Simplified the proof of a known result in group theory

Enhanced understanding of automorphism sums in finite abelian p-groups

Abstract

Using combinatorial techniques, we prove that the weighted sum of the inverse number of automorphisms of all finite abelian $p$-groups $\sum_G |G|^{-u} |\text{Aut}(G)|^{-1}$ is equal to $\prod_{j=u+1}^\infty\left(1-1/p^j\right)^{-1}$, where $u$ is a non-negative integer. This result was originally obtained by H. Cohen and H. W. Lenstra, Jr. In this paper we give a new elementary proof of their result.