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arXiv · 1407.3066

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

Pritam Majumder
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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.

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Pritam Majumder. 2014-07-11. An elementary proof of a power series identity for the weighted sum of all finite abelian p-groups. https://arxiv.org/abs/1407.3066

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