Combinatorial properties of intervals in finite solvable groups
type: Item
instance of: scholarly article
title: Combinatorial properties of intervals in finite solvable groups
zbMATH Open document ID: 0831.20022
author: Volkmar Welker
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-02-06T00:00:00Z
review text: Given a finite solvable group \(G\) and a subgroup \(H \leq G\), the interval \([H,G] = \{U \mid H \leq U \leq G\}\) in the subgroup poset is the object of interest in this quite penetrating paper in which the author has successfully introduced important combinatorial principles which combined with suitable group-theoretical results provide quite powerful means of analysis leading to a substantial collection of new results as well as to alternative proofs of several results in the area which are well-known there. The approach used depends on the fact that if \(\mu[H,G]\) denotes the Möbius number of the interval and if \(k(G)\) denotes the poset of all subgroups \(H\) of \(G\) such that \(\mu[H,G] = 0\), then \(\mu[H,G] = \mu[H,G]_k\), when \([H,G]_k = (k(G) \cap [H,G]) \cup \{G\}\). Furthermore, if \(\mu[H,G] = 0\), then a considerable effort is sufficient to show that \([H,G]_k\) is a lexicographically shellable poset whose shelling is determined and then permits determination of the Möbius number in terms of the numbers of maximal subgroups in the interval. Hence, e.g., a proof plus further insight into a vintage result of Gaschütz is provided. From \([H,G]\) and \([H,G]_k\) passage is made to \([H,G]/G\) and \([H,G]_k/G\) whose elements consist of conjugacy classes \([U] = \{U^g\mid g \in G\}\) of subgroups with \([U] \leq [V]\) iff \(U^g \leq V\) for some \(g \in G\). The approach already used is employed to determine the Möbius number \(\mu([H,G]_k / G)\) in terms of conjugacy classes of maximal subgroups, thence passing to \(\mu([H,G]/G)\) by equality. Knowledge of the length of orbits in this setting produces the important formula \(|G : N_G(H) |\mu[H,G] = |HG : H|\mu([H,G]/G)\) which is analyzed and refined further to obtain several consequences of interest also.
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 757790
zbMATH Keywords: complemented lattices
zbMATH Keywords: intervals in subgroup poset
zbMATH Keywords: conjugacy classes of subgroups
zbMATH Keywords: finite solvable groups
zbMATH Keywords: Möbius number
zbMATH Keywords: lexicographically shellable posets
zbMATH Keywords: numbers of maximal subgroups
zbMATH Keywords: conjugacy classes of maximal subgroups
zbMATH Keywords: length of orbits
reviewed by: J. Neggers
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072358
OpenAlex ID: W2010560666
cites work: Shellable and Cohen-Macaulay Partially Ordered Sets
cites work: Die Eulersche Funktion endlicher auflösbarer Gruppen
cites work: Pre-Frattini groups.
cites work: Q5577154
cites work: A lattice theoretic characterization of prefrattini subgroups
cites work: On the Möbius function of a finite group
cites work: Endliche Gruppen I
cites work: Type d'homotopie des treillis et treillis des sous-groupes d'un groupe fini. (Homotopy type of lattices and lattices of subgroups of a finite groups)
cites work: Die Praefrattinigruppe im Intervall eines Untergruppenverbandes. (The prefrattini group in the interval of a subgroup lattice)
cites work: On the foundations of combinatorial theory I. Theory of M�bius Functions
cites work: Q3748279
cites work: Q3232305
cites work: The combinatorics and the homology of the poset of subgroups of \(p\)-power index
cites work: The poset of conjugacy classes of subgroups in a finite solvable group
cites work: Equivariant homotopy of posets and some applications to subgroup lattices
cites work: On the poset of partitions of an integer
DOI: 10.1216/RMJM/1181072358
Recommended article: Intersecting subgroups of finite soluble groups
Recommended article: Intervals in subgroup lattices of finite groups
Recommended article: Q4891339
Recommended article: Solvability of finite groups
Recommended article: On Boolean intervals of finite groups
Recommended article: A combinatorial property of certain infinite groups
Recommended article: Q4735340
Recommended article: Q4217967
Recommended article: Q4840265
Recommended article: On intervals in subgroup lattices of finite groups
description: scientific article; zbMATH DE number 757790