Eliminating components in Quillen's Conjecture

On Quillen's conjecture for p-solvable groups

We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without employing it.

Acyclic $2$-dimensional complexes and Quillen's conjecture

Let $G$ be a finite group and $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $\mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)\neq 1$ for any group $G$ admitting a $G$-invariant acyclic $p$-subgroup complex of dimension $2$. In particular, it follows that Quillen's conjecture holds for groups of $p$-rank $3$. We also apply this result to establish Quillen's conjecture for some particular groups not considered in the seminal work of Aschbacher--Smith.

Quillen的猜想与单位ary群

Quillen's conjecture and unitary groups

研究证明简单单位ary群的p-扩展的Quillen位势在最大可能维度有非零同调，验证了Aschbacher-Smith的猜想，并支持Quillen关于p-子群位势的猜想。

特征p下的分支与Quillen猜想

Components in characteristic $p$ and Quillen's conjecture

该研究针对群论中的Quillen猜想，在温和归纳假设下证明含特征p李型单群分支且O_p(G)=1的群，其Quillen偏序集有非零有理同调，为p=2时的猜想研究提供关键依据。

An approach to Quillen's conjecture via centralizers of simple groups

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…

A geometric approach to Quillen's conjecture

We introduce {\em admissible collections} for a finite group $G$ and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the {\em Quillen dimension at $p$ property}, a strong version of Quillen's conjecture, at a given odd prime divisor $p$ of $|G|$. Compared to the methods in \cite{AS1993}, our techniques are simpler.