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.
Citation
Nobuo IIYORI. Masato SAWABE. "Homology of the Complex of All Non-trivial Nilpotent Subgroups of a Finite Non-solvable Group." Tokyo J. Math. 42 (1) 113 - 120, June 2019. https://doi.org/10.3836/tjm/1502179264