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I am wondering if there is any literature on general formula of the Moebius function of subgroup lattices of any finite abelian group $G$? What I know is

When $G$ is cyclic, the Moebius function is simply the classical number theoretic one.

When $G=(\mathbb Z/p\mathbb Z)^r$, the formula involves the number of $k$-dimensional linear subspace of $G$.

But is there some formula for any finite abelian group?

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    $\begingroup$ Wouldn't G-C Rota, "On the foundations of combinatorial theory I. Theory of Möbius Functions", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 2, pp. 340–368 (1964), be the sort of thing you are looking for? $\endgroup$ Dec 5 '21 at 9:00
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Since every interval of the subgroup lattice $\mathcal{L}(G)$ of a finite abelian group $G$ is isomorphic to the subgroup lattice of some finite abelian group, we can restrict ourselves to $\mu(\hat{0},\hat{1})$, where $\hat{0}$ is the bottom element (the trivial subgroup of $G$) and $\hat{1}$ is the top element (the group $G$ itself) of $\mathcal{L}(G)$. If $G$ and $H$ have relatively prime orders, then $\mathcal{L}(G\times H)= \mathcal{L}(G) \times \mathcal{L}(H)$. Thus (EC1, second ed., Proposition 3.8.2) $\mu_{G\times H}(\hat{0},\hat{1}) = \mu_G(\hat{0},\hat{1})\mu_H(\hat{0},\hat{1})$, so we can assume that $G$ has prime power order $p^n$. Then $\mathcal{L}(G)$ is atomic (i.e., $\hat{1}$ is a join of atoms, or $G$ is generated by the subgroups of order $p$) if and only if $G$ is elementary abelian (a product of groups of order $p$). Since for any finite lattice $L$, $\mu(\hat{0},\hat{1})\neq 0$ implies that $L$ is atomic (EC1, Corollary 3.9.5), we have $\mu(\hat{0},\hat{1})=0$ unless $G$ is elementary abelian. Finally, if $G$ is elementary abelian of order $p^n$, then it is well-known (e.g., EC1, equation (3.34)) that $\mu(\hat{0},\hat{1})=(-1)^n p^{{n\choose 2}}$.

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We can say something reasonably general about finite abelian groups.

Throughout, we'll let $\mathcal{L}(G)$ be the lattice of subgroups of a finite group $G$ and $\overline{\mathcal{L}(G)}$ be the proper part of $\mathcal{L}(G)$ (i.e., $\mathcal{L}(G)$ without the top and bottom elements). Recall that the order complex $\Delta(P)$ of a poset $P$ is the (abstract) simplicial complex whose faces are the chains of $P$.

The following result is due to Kratzer and Thévanaz (Corollaire 4.10). Translation due to John Shareshian.

Theorem: Let $G$ be a finite solvable group with chief series $$1 = G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \cdots \triangleleft G_r = G.$$ For $1 \leq i < r$, let $m_i$ be the number of complements to $G_i/G_{i-1}$ in $G/G_{i-1}$. Then $\Delta(\overline{\mathcal{L}(G)})$ has the homotopy type of a wedge of $$m = \displaystyle\prod_{i=1}^{r-1}m_i$$ $(r-2)$-spheres.

On the other hand, we have one of the foundational results of poset topology, Philip Hall's Theorem. See, e.g., Wachs' notes on poset topology.

Philip Hall's Theorem: For any poset $P$ with top element $\hat{1}$ and bottom element $\hat{0}$, $$\mu_P(\hat{0},\hat{1}) = \tilde{\chi}(\Delta(\overline{P})),$$ where $\tilde{\chi}$ is the reduced Euler characteristic and $\overline{P}$ is $P\setminus \{\hat{0},\hat{1}\}$.

In particular, when $G$ is abelian (and hence solvable), we can compute the Möbius function $\mu_{\mathcal{L}(G)}(K,H)$ by applying Kratzer and Thévanaz's result to the interval $[K,H]$ in $\mathcal{L}(G)$.

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