On the Reduced Lefschetz Module and the Centric $p$-Radical Subgroups

Abstract

The purpose of this paper is to show that the reduced Lefschetz module of the $G$-poset $\mathcal{B}_{p}^{cen}(G)$ consisting of all centric $p$-radical subgroups of a finite group $G$ is an $\mathcal{X}$-projective virtual $\mathbb{Z}_{p}[G]$-module where $\mathcal{X}$ is a family of $p$-subgroups of the normalizers of non-centric $p$-radical subgroups of $G$. As corollary, we have a lower bound of the $p$-power of the reduced Euler characteristic $\tilde{\chi}(\mathcal{B}_{p}^{cen}(G))$.