A note on the decomposition of the Burnside rings of finite groups
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 \(\mathfrak X\).
scientific article; zbMATH DE number 872589