Character polynomials and the Möbius function
type: Item
instance of: scholarly article
title: Character polynomials and the Möbius function
zbMATH Open document ID: 0841.20005
DOI: 10.1007/BF01270687
author: Herbert Pahlings
published in: Archiv der Mathematik
publication date: 1996-07-14T00:00:00Z
review text: This elegant paper concerns the Möbius function \(\mu\) on the partially ordered set of all subgroups of a finite group \(G\). Assume \(G\) is acting by permutations on some set \(\Omega\). Then the permutation structure of the action of \(G\) on the permutation modules \(\mathbb{F}_q \Omega\) over \(\mathbb{F}_q\) can be described by some polynomials in \(q\) [see \textit{H. Pahlings} and \textit{W. Plesken}, J. Reine Angew. Math. 380, 178-195 (1987; Zbl 0613.20008), or \textit{A. Turull}, Proc. Am. Math. Soc. 103, No. 3, 463-467 (1988; Zbl 0651.20012)]. The present paper uses these polynomials to obtain relations among the values of the \(\mu\) function on the transitive subgroups of \(G\). Particular attention is paid to the symmetric and alternating groups. It is proved, using the classification of finite simple groups, that if \(p\) is any prime, then \(\mu(1,S_p)=(-1)^{p+1}p!/2\). For many such \(p\), it is also shown that \(\mu(1,A_p)=-\mu(1,S_p)\).
reviewed by: Alexandre Turull
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 800592
zbMATH Keywords: Möbius function
zbMATH Keywords: partially ordered sets
zbMATH Keywords: finite groups
zbMATH Keywords: actions
zbMATH Keywords: permutation modules
zbMATH Keywords: transitive subgroups
zbMATH Keywords: symmetric and alternating groups
describes a project that uses: GAP
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01270687
OpenAlex ID: W2092699541
description: scientific article; zbMATH DE number 800592