The generalized Burnside ring with respect to p-centric subgroups

Let X be the set of all p-centric subgroups of a finite group G and a prime p. This paper shows that the certain submodule Ω(G,X)(p) of the Burnside ring Ω(G)(p) of G over the localization Z(p) of Z at p has a unique ring structure such that the mark homomorphism φ(p) relative to X is an injective homomorphism. A key lemma of this paper is that X satisfies the condition (C)p that is discussed by [T. Yoshida, The generalized Burnside ring of a finite group, Hokkaido Math. J. 19 (1990) 509–574]. Díaz and Libman showed that certain ring Ap-cent(G)(p) is isomorphic to the Burnside ring of the fusion system associated to G and a Sylow p-subgroup in [A. Díaz, A. Libman, The Burnside ring of fusion systems, preprint, 2007]. This paper shows that Ap-cent(G)(p) is isomorphic to Ω(G,X)(p).