Using combinatorial techniques, we prove that the weighted sum of the inversenumber 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-negativeinteger. 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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机译：使用组合技术，我们证明了所有有限阿贝尔$ p $-组$ \ sum_G | G | ^ {-u} | \ text {Aut}（G）| ^ {-1}的自同构逆数的加权和$等于$ \ prod_ {j = u + 1} ^ \ infty \ left（1-1 / p ^ j \ right）^ {-1} $，其中$ u $是非负整数。此结果最初是由H. Cohen和H. W. Lenstra，Jr.获得的。在本文中，我们给出了其结果的新的基本证明。
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