Homology of the Complex of All Non-trivial Nilpotent Subgroups of a Finite Non-solvable Group

Abstract

Let $G$ be a finite non-solvable group. We study homology of the complex $\mathcal{N}(G)$ of all non-trivial nilpotent subgroups of $G$. The determination of $H_{n}(\mathcal{N}(G))$ is reduced to that of homology of its subcomplex $\mathcal{N}_{\pi_{1}}(G)$ consisting of all nilpotent $\pi_{1}$-subgroups, where $\pi_{1}$ is the connected component of the prime graph of $G$ containing 2. Furthermore, $\mathcal{N}_{\pi_{1}}(G)$ is connected if $G$ possesses no strongly embedded subgroups.