论文标题
通过简单组的集中化器来猜测Quillen的一种方法
An approach to Quillen's conjecture via centralizers of simple groups
论文作者
论文摘要
我们表明，对于有限的组$ g $的任何给定的子组$ h $，Quillen Poset $ \ Mathcal {a} _p（g）$ abelian $ p $ -subGroups的$ p $ -subgroups是从$ h $ h $ h $ h $ h中获得的。我们使用这个想法来研究Quillen的猜想，该猜想断言，如果$ \ Mathcal {a} _p（g）$是合可能的，那么$ g $具有非平凡的普通$ p $ -subgroup。我们证明，原始猜想等于猜想的$ \ mathbb {z} $ - cosyclic版本（通过通过$ \ mathbb {z} $ - acyclic替换来获得。我们还使用$ \ mathbb {q} $ - acyclic（强）版本的猜想，将其研究减少为简单订单的直接产品的扩展，可将其排除在$ p $和$ p $ rank中，至少$ 2 $。这允许扩展Aschbacher-Smith的结果，并为$ p $ lank的组建立强有力的猜想，最多为$ 4 $。
We show that, 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 centralizers in $H$. We use 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 order divisible by $p$ and $p$-rank at least $2$. This allows to extend results of Aschbacher-Smith and to establish the strong conjecture for groups of $p$-rank at most $4$.
