The coset poset and probabilistic zeta function of a finite group
For each finite group \(G\) and each integer \(s\geq 1\), let \(P(G,s)\) denote the probability that \(s\) elements chosen at random from \(G\) generate all of \(G\). A theorem of \textit{P. Hall} [see Q. J. Math., Oxf. Ser. 7, 134-151 (1936; Zbl 0014.10402)] shows that \(P(G,s)=\sum\mu(G,H)|G:H|^{-s}\) where the sum is over all subgroups \(H\) of \(G\) and \(\mu\) is a generalized Möbius function. This sum is clearly defined for all real \(s\) and its reciprocal is sometimes called the zeta function for \(G\). The current paper studies the value of the integer \(P(G,-1)\) which turns out to have a topological interpretation. Consider the ``coset poset'' \(\mathcal{C}(G)\) defined as the set of cosets of all proper subgroups of \(G\) ordered by inclusion. Then the simplicial complex \(\Delta(\mathcal{C}(G))\) consisting of the finite chains in \(\mathcal{C}(G)\) has Euler characteristic \(1-P(G,-1)\). Using this topological connection the author shows that every prime \(p\) dividing \(|G|\) also divides \(P(G,-1)\), and he gives group theoretic conditions which imply that \(P(G,-1)\) is divisible by higher powers of \(p\). Apart from this connection with \(P(G,-1)\) there are many other open questions concerning the relationship between topological properties of \(\mathcal{C}(G)\) and properties of \(G\).
scientific article; zbMATH DE number 1429480
scientific article