The combinatorics and the homology of the poset of subgroups of \(p\)-power index
type: Item
instance of: scholarly article
title: The combinatorics and the homology of the poset of subgroups of \(p\)-power index
zbMATH Open document ID: 0812.20006
DOI: 10.1016/0022-4049(93)90044-T
author: Michael Weidner
author: Volkmar Welker
published in: Journal of Pure and Applied Algebra
publication date: 1995-05-09T00:00:00Z
review text: In this quite multidimensional paper, the objects of interest are posets \(S^ p(G)\) of all subgroups \(H \neq G\) (a finite group) of prime power index (or generalizations to \(S^ \pi (G)\), \(\pi\) a set of primes), studied mainly through considerations of the order complex \(CS^ p(G)\) of \(S^ p(G)\) and its geometric realization \(| CS^ p(G)|\), which is shown to be a homotopy wedge of spheres of a fixed (interesting) dimension. Hence also the Möbius number \(\mu(S^ p(G))\) is obtained, thereby providing means to represent \(G\) on \(CS^ p(G)\) which in turn permits a representation of \(G\) realized as a set of matrices with entries in \(\{-1, 0, +1\}\). The authors draw on a wealth of results from the theory of finite groups along with the algebraic-combinatorial theory involved in the shellability theory of posets and their complexes as well as their own earlier results of a related nature, some dealing with subgroup posets different from those discussed here. Significant by-products include a proof of a dual of a conjecture of Quillen (i.e., \(S^ p(G)\) is \(G\)-contractible iff contractible iff \(\mu(S^ p(G)) = 0\) iff \(O^ p(G)\), the minimal normal subgroup of \(p\)- power index, \(\neq G\)) and a conjecture that \(S^ p_ c(G)\), a closely associated subposet of \(S^ p(G)\), \(G\)-homotopy equivalent to \(S^ p(G)\), is CL-shellable for all finite groups \(G\), based on observations made in arguments in the text of the paper. From the proofs and statements of the results, it is clear that ``comparability'' is possibly as important a property as actual ``order'' once the group \(G\) and the prime \(p\) have been fixed. En fin the paper makes a contribution both to the theory of finite groups and to the topological combinatorics of posets.
reviewed by: J. Neggers
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 05E25
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 55Q20
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 500586
zbMATH Keywords: homotopy type
zbMATH Keywords: homology
zbMATH Keywords: homotopy equivalent
zbMATH Keywords: prime power index
zbMATH Keywords: order complex
zbMATH Keywords: geometric realization
zbMATH Keywords: homotopy wedge of spheres
zbMATH Keywords: Möbius number
zbMATH Keywords: finite groups
zbMATH Keywords: shellability theory of posets
zbMATH Keywords: subgroup posets
zbMATH Keywords: CL-shellable
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 500586