The inverse limit of the Burnside ring for a family of subgroups of a finite group
Let \(A(G)\) the Burnside ring of a finite nontrivial group \(G\). If \(\mathcal{F}\subseteq \mathcal{S}(G)\) is a set of subgroups of \(G\) closed under subgroups and conjugation, then \(\mathfrak{F}\) denotes the category with objects the elements of \(\mathcal{F}\), and morphisms are triples \((H,g,K)\) such that \(H,K \in \mathcal{F}\) and \(g \in G\) with \(gHg^{-1}\subseteq K\). The ring \(A(\mathfrak{F})\) is obtained by taking the inverse limit of \(A(H)\), where \(H \in F\), and let \(\mathrm{res}^G_F : A(G) \to A(F)\) be the restriction homomorphism. The main result in the paper says that if \(G\) is nilpotent group, and \(\mathcal{F} = \mathcal{S}(G)\setminus\{G\}\), then \(\mathrm{Im}(\mathrm{res}^G_{\mathcal{F}})\) coincides with \(A(\mathfrak{F})\) if and only if G is a cyclic group of which the order is a prime or a product of distinct primes.
scientific article; zbMATH DE number 6901713