A weighted Möbius function

Fix an odd prime ℓ and let G be the poset of isomorphism classes of finite abelian ℓ-groups, ordered by inclusion. If ξ:G→R≥0 is a discrete probability distribution on G and A∈G, define the Ath moment of ξ to be ∑B∈G|Surj(B,A)|ξ(B). The question of determining conditions that ensure ξ is completely determined by its moments has been of recent interest in many problems of Cohen-Lenstra type. Furthermore, recovering ξ from its moments requires a new Möbius-type inversion formula on G. In this paper, we define this function, relate it to the classical Möbius function on the poset of subgroups of a fixed group, and prove two theorems about when it vanishes. As one corollary of these theorems, we obtain an analog of Hall's theorem on the vanishing of the classical Möbius function. As another, we obtain an infinite family of pairs of groups on which the classical Möbius function vanishes; obtaining such pairs is a group-theoretic topic of recent interest.