Rational homology and Quillen's conjecture

We prove Quillen's conjecture for all finite groups and all primes. More precisely, if a finite group G has trivial largest normal p-subgroup $O_p(G)$, then the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology. This establishes the stronger rational-homology form of the conjecture.