Generalized Burnside rings and group cohomology
Let \(G\) be a finite group, \(X\) a finite \(G\)-set, \(M\) a \({\mathbb Z}G\)-module and \(H_X^*(G,M)\) the cohomology of \(G\) associated to \(X\) with coefficients in \(M\). A positioned \(G\)-set is a pair of the form \((X,u)\), where \(X\) is a \(G\)-set and \(u\) is a class in \(H_X^n(G,M)\). The set of isomorphism classes of positioned \(G\)-sets form a semi-ring and the cohomological Burnside ring \(B^n(G,M)\) is defined as the Grothendieck ring of this semi-ring. This is a generalization of so called crossed Burnside ring and the monomial Burnside ring. Let \(Cl(G)\) be the set of conjugacy classes of subgroups of \(G\) and let \(W_G(H)=N_G(H)/H\) be the Weyl group of \(H\) in \(G\). The ghost ring of the cohomological Burnside ring is \(\beta^n(G,M)=\oplus_{[H]\in Cl(G)} ({\mathbb Z}H^n(H,M))^{W_G(H)}.\) Then there exists an exact sequence of abelian groups \[ 0\rightarrow B^n(G,M)\rightarrow\beta^n(G,M)\rightarrow Obs^n(G,M)\rightarrow 0, \] where \(Obs^n(G,M)=\oplus_{[H]\in Cl(G)} E_H,\) \(E_H={\hat H}^0(W_G(H),{\mathbb Z}H^n(H,M))\) being the Tate group (this is the fundamental theorem for cohomological Burnside rings).
scientific article; zbMATH DE number 5143570