Partially ordered sets of non-trivial nilpotent $\pi$-subgroups

Abstract

In this paper, we introduce a subposet $\mathcal{L}_{\pi}(G)$ of a poset $\mathcal{N}_{\pi}(G)$ of all non-trivial nilpotent $\pi$-subgroups of a finite group $G$. We examine basic properties of subgroups in $\mathcal{L}_{\pi}(G)$ which contain the notion of both radical $p$-subgroups and centric $p$-subgroups of $G$. It is shown that $\mathcal{L}_{\pi}(G)$ is homotopy equivalent to $\mathcal{N}_{\pi}(G)$. As examples, we investigate in detail the case where symmetric groups.