The coset poset and probabilistic zeta function of a finite group
type: Item
instance of: scholarly article
title: The coset poset and probabilistic zeta function of a finite group
zbMATH Open document ID: 0973.20016
published in: Journal of Algebra
publication date: 2001-08-28T00:00:00Z
review text: 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 \(G\).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 20P05
zbMATH DE Number: 1429480
zbMATH Keywords: finite groups
zbMATH Keywords: zeta functions
zbMATH Keywords: generalized Möbius functions
zbMATH Keywords: subgroups
zbMATH Keywords: cosets
zbMATH Keywords: Euler characteristics
author: Kenneth S. Brown
describes a project that uses: GAP
MaRDI profile type: Publication
OpenAlex ID: W2017827867
cites work: Sylow intersections and fusion
cites work: Q4320289
cites work: Simple connectivity of \(p\)-group complexes
cites work: Alperin's fusion theorems and G-posets
cites work: Q4871782
cites work: Q4885303
cites work: Q3681031
cites work: Set functors equipped with a double action
cites work: Polynomial ideals and classes of finite groups
cites work: Euler characteristics of groups: The \(p\)-fractional part
cites work: A generalization of Sylow's third theorem
cites work: Q5848209
cites work: Q3251484
cites work: Zu einem von B. H. und H. Neumann gestellten Problem
cites work: Die Eulersche Funktion endlicher auflösbarer Gruppen
cites work: THE EULERIAN FUNCTIONS OF A GROUP
cites work: Endliche Gruppen I
cites work: Discrete Series of GLn Over a Finite Field. (AM-81)
cites work: Positively finitely generated groups
cites work: Homotopy properties of the poset of nontrivial p-subgroups of a group
cites work: On the shellability of the order complex of the subgroup lattice of a finite group
cites work: Q5680302
cites work: The affine group. I: Bruhat decomposition
cites work: Q3926778
cites work: Maximal subgroups of direct products
DOI: 10.1006/JABR.1999.8221
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description: scientific article; zbMATH DE number 1429480