For any given subgroup H of a finite group G, the Quillen poset ${\mathcal {A}}_p(G)$ of nontrivial elementary abelian p-subgroups is obtained from ${\mathcal {A}}_p(H)$ by attaching elements via their centralisers in H. We exploit this idea to study Quillen’s conjecture, which asserts that if ${\mathcal {A}}_p(G)$ is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the ${{\mathbb {Z}}}$ -acyclic version of the conjecture (obtained by replacing ‘contractible’ by ‘ ${{\mathbb {Z}}}$ -acyclic’). We also work with the ${\mathbb {Q}}$ -acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least $2$ . This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most $4$ .

中文翻译：

---

一种通过简单群的集中器来解决奎伦猜想的方法

对于任何给定的子组H有限群的G, 奎伦派塞特 ${\mathcal {A}}_p(G)$ 非平凡初等阿贝尔的p-subgroups 来自 ${\mathcal {A}}_p(H)$ 通过它们的扶正器附加元素H. 我们利用这个想法来研究奎伦的猜想，它断言如果 ${\mathcal {A}}_p(G)$ 那么是可收缩的G有一个非平凡的正常p-亚组。我们证明原猜想等价于 ${{\mathbb {Z}}}$ - 猜想的非循环版本（通过将 'contractible' 替换为 ' ${{\mathbb {Z}}}$ -无环'）。我们还与 ${\mathbb {Q}}$ -该猜想的无环（强）版本，将其研究简化为简单群的直接乘积的扩展p- 至少排名 $2$ . 这使我们能够扩展 Aschbacher 和 Smith 的结果，并为p- 最多排名 $4$ .