A note on the decomposition of the Burnside rings of finite groups
type: Item
instance of: scholarly article
title: A note on the decomposition of the Burnside rings of finite groups
zbMATH Open document ID: 0849.20003
author: Fumihito Oda
published in: Hokkaido Mathematical Journal
publication date: 1996-11-05T00:00:00Z
review text: One of the main results in a paper by \textit{J. Thévenaz} [J. Lond. Math. Soc., II. Ser. 45, No. 3, 446-468 (1992; Zbl 0833.20016)] is a decomposition \(\Omega(G)=\Omega(G,{\mathfrak X})\oplus K(G,{\mathfrak X})\) of the Burnside ring \(\Omega(G)\) of a finite group \(G\) into subgroups \(\Omega(G,{\mathfrak X})\) and \(K(G,{\mathfrak X})\). Here \(\mathfrak X\) is the set of \(p\)-local subgroups of \(G\) for a prime \(p\), \(\Omega(G,{\mathfrak X})\) is the span of all canonical basis elements in \(\Omega(G)\) corresponding to subgroups in \(\mathfrak X\), and \(K(G, {\mathfrak X})\) is the ideal of all elements in \(\Omega(G)\) mapped to 0 under the fixed point homomorphisms \(\Omega(G)\to\mathbb{Z}\) corresponding to subgroups in \(\mathfrak X\). The proof by Thévenaz involves the simplicial complex of chains of nontrivial \(p\)-subgroups of \(G\). In the paper under review the author notes that Thévenaz's result follows easily from more general results of \textit{T. Yoshida} [Hokkaido Math. J. 19, No. 3, 509-574 (1990; Zbl 0731.20002)]. The main theorem is a decomposition \(\Omega(G)=\Omega(G,{\mathfrak X})\oplus K(G,{\mathfrak X})\) where \(\mathfrak X\) now is an arbitrary family of subgroups of \(G\) closed under conjugation and satisfying the following condition: \(S\in{\mathfrak X}\), \(g\in N_G(S)\Rightarrow\langle g\rangle S\in{\mathfrak X}\). There are similar decompositions for localized Burnside rings with respect to more general families \(\mathfrak X\).
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 872589
zbMATH Keywords: Alperin's conjecture
zbMATH Keywords: decompositions
zbMATH Keywords: Burnside rings
zbMATH Keywords: finite groups
zbMATH Keywords: \(p\)-local subgroups
zbMATH Keywords: localized Burnside rings
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1351516710
OpenAlex ID: W2003968859
DOI: 10.14492/HOKMJ/1351516710
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description: scientific article; zbMATH DE number 872589