The generalized Burnside rings with respect to a collection of self-normalizing subgroups

In this paper, we study the generalized Burnside ring Ω(G,D) with respect to a collection D of self-normalizing subgroups. It is shown that the ordinary Burnside ring Ω(G) can be decomposed into Ω(G,D) and the kernel of a certain ring homomorphism ρD. A basis of is also investigated. Furthermore we give a formula for the unit of Ω(G,D), which is related with the Euler characteristic. As example, we take D as a collection of the normalizers of certain p-radical subgroups of G. Then the unit is realized as the Lefschetz invariant of the order complex of D.