Author : Fumihito Oda

Title  : The generalized Burnside ring with respect to $p$-centric subgroups

Abstruct : Let ${\mathfrak X}$ be the set of all $p$-centric subgroups
of a finite group
$G$ and a prime $p.$ This paper shows that  the certain submodule
$\Omega(G,{\mathfrak X})_{(p)}$ of the Burnside ring $\Omega(G)_{(p)}$ of
$G$ over the localization ${\mathbb{Z}}_{(p)}$ of ${\mathbb{Z}}$ at $p$ has
a unique ring structure such that the mark homomorphism
$\varphi_{(p)}$ relative to
${\mathfrak X}$ is an injective homomorphism. A key lemma of this paper is that
${\mathfrak X}$ satisfies the condition $({\bf{C}})_p$ that is
discussed by [Yo90].
D\'iaz and Libman showed that certain ring $\mathcal{A}^{p\mbox{{\tiny
-cent}}}(G)_{(p)}$
  is isomorphic to the Burnside ring of the  fusion system associated to
$G$ and a Sylow $p$-subgroup in [DL07]. This paper shows that
 $\mathcal{A}^{p\mbox{{\tiny -cent}}}(G)_{(p)}$ is isomorphic to
 $\Omega(G,{\mathfrak X})_{(p)}.$

Status : Preprint.