How do I go about proving Quillen's Homotopy Equivalence of Chevalley (subgroups of) automorphism groups of finite classical Lie algebras. Given $\sigma$ , such that Chevalley’s construction $GU$$n$($q$) is obtained from $GL$$n$($q^2$) by twisting. 


$\sigma$ defines an automorphism ($a\_{i,j}$) $\rightarrow$ ($a^{q}\_{i,j})^{−tr}$ of $GL\_n(q^2)$.
Then $GU\_n(q)$ = $GL\_n(q^2)^\sigma$ 


The proof in and out of itself should enforce the existance of posets $S\_p(G)$ consisting of non-identity p-subgroups of a finite group $G$, which are homotopy equivalent to the subposet $A\_p(G)$
consisting of non-identity elementary abelian p-subgroups.



Homotopy Equivalence of Posets for the Weyl Group
The homotopy equivalence of the two posets is discussed in this question [Status of Quillen's conjecture on elementary abelian p-groups](https://mathoverflow.net/questions/38890/status-of-quillens-conjecture-on-elementary-abelian-p-groups) and its answers. The first part of the question I maintain makes little sense and needs to be reformulated. 


