Numbers of classes and chains of subgroups in finite groups
type: Item
instance of: scholarly article
title: Numbers of classes and chains of subgroups in finite groups
zbMATH Open document ID: 0907.20031
author: I. M. Isaacs
published in: Journal of Algebra
publication date: 1999-03-07T00:00:00Z
review text: Let \(X\) be a normal subset of a finite group \(G\), and let \(m_N(V)\) denote the number of conjugacy classes of \(N\) contained in \(V\cap X\), for subgroups \(V\subseteq N\subseteq N_G(V)\subseteq G\). Moreover, let \(\mathcal F\) be a conjugation-invariant collection of subgroups of \(G\) such that each element \(x\in X\) is contained in a unique smallest member \(F_x\in{\mathcal F}\). Finally, let \(\mathcal S\) denote the \(G\)-poset of nonempty chains in \(\mathcal F\), and let \(G_\sigma\) denote the stabilizer and \(V_\sigma\) the smallest member of a chain \(\sigma\in{\mathcal S}\). The author shows that \(m_G(G)=\sum_{\sigma\in{\mathcal S}}(-1)^{|\sigma|+1} m_{G_\sigma}(V_\sigma)/| G:G_\sigma|\). This, together with a related theorem, generalizes earlier results by \textit{G. Robinson} and this reviewer [e.g., J. Algebra 205, No. 1, 326-332 (1998)]. The proofs are elementary and simplify the earlier representation-theoretic approach.
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1213844
zbMATH Keywords: Möbius function
zbMATH Keywords: numbers of conjugacy classes
zbMATH Keywords: posets
zbMATH Keywords: chains
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: jabr.1998.7430
OpenAlex ID: W2150346619
cites work: On the Möbius function of a finite group
cites work: On local control of the number of conjugacy classes
DOI: 10.1006/JABR.1998.7430
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description: scientific article; zbMATH DE number 1213844