The generalized Burnside ring with respect to \(p\)-centric subgroups
Let \(G\) be a finite group and \(p\) a prime number. Denote by \({\mathbb Z}_{(p)}\) the localization of \(\mathbb Z\) at \(p\), and for a \(\mathbb Z\)-module \(M\), let \(M_{(p)}:= {\mathbb Z}_{(p)}\otimes_{\mathbb Z}M\). Let \(\mathfrak X\) be the set of \(p\)-centric subgroups of \(G\). The author proves that \(\mathfrak X\) satisfies the following condition (\textbf{C})\(_p\) introduced by \textit{T.~Yoshida} [Hokkaido Math. J. 19, 509--574 (1990; Zbl 0731.20002)]. If \(S\in \mathfrak X\) and \(gS\) belongs to a Sylow \(p\)-subgroup of \(N_G(S)/S\), then \(\overline{\langle g\rangle S}\in \mathfrak X\), where \(\overline H:=\{ S\in \mathfrak X\mid H\leq S\}\). Now let \(\Omega(G,\mathfrak X)\) be the submodule of the ordinary Burnside ring \(\Omega(G)\) generated by the elements \([G/H]\), for \(H\in \mathfrak X\). This is endowed by a certain mark homomorphism \(\phi\). Then it is shown that \(\Omega(G,\mathfrak X)_{(p)}\) has a unique ring structure such that the mark homomorphism \(\phi_{(p)}\) is an injective ring homomorphism. Finally, the author shows that \(\Omega(G,\mathfrak X)_{(p)}\) is isomorphic to a certain ring \({\mathcal A}^{p{\text{-cent}}}(G)_{(p)}\) recently introduced by \textit{A.~Díaz} and \textit{A.~Libman}, and draws conclusions concerning the fusion system associated to \(G\) and a Sylow \(p\)-subgroup \(S\).
scientific article; zbMATH DE number 5380149
scientific article