Fonction de Möbius d'un groupe fini et anneau de Burnside. (Möbius function of a finite group and the Burnside ring)
type: Item
instance of: scholarly article
title: Fonction de Möbius d'un groupe fini et anneau de Burnside. (Möbius function of a finite group and the Burnside ring)
zbMATH Open document ID: 0592.20004
DOI: 10.1007/BF02566359
author: Charles Kratzer
author: Jacques Thévenaz
published in: Commentarii Mathematici Helvetici
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/139984
review text: This paper gives various relationships between the structure and idempotents of the Burnside ring \(\Omega\) (G) of a finite group G and its lattice S(G) of subgroups, in particular the Möbius function of S(G). Crapo's formula for the Möbius function of a lattice in terms of the complements of a fixed point of the lattice is reproved for S(G). The Möbius function for nilpotent and soluble groups is set down explicitly, the latter in terms of the number of complements of the terms of a principal series for the group. If \(H\leq G\), and if \(| G:G'|_ o\) is the product of the distinct prime factors of \(| G:G'|\), then \(| G:G'|_ o\cdot \mu (H,G)\) is a multiple of \(| N_ G(H):H|\); and this last result is best possible.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 3950795
zbMATH Keywords: lattice of subgroups
zbMATH Keywords: idempotents
zbMATH Keywords: Burnside ring
zbMATH Keywords: Möbius function
zbMATH Keywords: Crapo's formula
zbMATH Keywords: soluble groups
zbMATH Keywords: principal series
MaRDI profile type: Publication
OpenAlex ID: W1569647734
description: scientific article; zbMATH DE number 3950795