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Let G be a finite group and Ap(G) be the poset of nontrivial elementary abelian p-subgroups of G. Quillen conjectured that Op(G) is nontrivial if Ap(G) is contractible. We prove that Op(G) = 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.

We investigate a stronger formulation of Webb's conjecture on the contractibilty of the orbit space of the p-subgroup complexes in terms of finite topological spaces. The original conjecture, which was first proved by Symonds and, more recently, by Bux, Libman and Linckelmann, can be restated in terms of the topology of certain finite spaces. We propose a stronger conjecture, and prove various particular cases by combining fusion theory of finite groups and homotopy theory of finite spaces.