Moebius function of finite abelian groups

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?

• 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? Dec 5 '21 at 9:00

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 $$\mu(\hat{0},\hat{1})=(-1)^n p^{{n\choose 2}}$$.

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 $$\mathcal{L}(G)$$.