An elementary proof of a power series identity for the weighted sum of all 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.

6 pages, submitted to Discrete Mathematics