Quillen's conjecture on the poset of elementary abelian p-subgroups

Theorem 1.1: for every finite group $G$ and prime $p$, if $O_p(G)=1$ then $\widetilde H_*(\mathcal A_p(G);\mathbb Q)\ne0$ (augmented, so the empty poset counts). This gives Quillen's conjecture and its stronger rational-homology form, and acyclicity over any field characterizes $O_p(G)\ne1$. Priority: the manuscript credits Diaz Ramos (arXiv:2303.15613v6, July 2026) with the rational form at all odd primes, so the new content is chiefly the prime $2$ (via Piterman's component eliminations) plus a direct proof of the unitary dimension property. It does not prove the Quillen dimension property (homology in top degree) in general.

Theorem 1.1: for every finite group $G$ and prime $p$, if $O_p(G)=1$ then $\widetilde H_*(\mathcal A_p(G);\mathbb Q)\ne0$ (augmented, so the empty poset counts). This gives Quillen's conjecture and its stronger rational-homology form, and acyclicity over any field characterizes $O_p(G)\ne1$. Priority: the manuscript credits Diaz Ramos (arXiv:2303.15613v6, July 2026) with the rational form at all odd primes, so the new content is chiefly the prime $2$ (via Piterman's component eliminations) plus a direct proof of the unitary dimension property. It does not prove the Quillen dimension property (homology in top degree) in general.