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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.

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Quillen's conjecture on the poset of elementary abelian p-subgroups · Quillen's conjecture

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Registry verification: unreviewed · announcement · candidate

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

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No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Quillen's Conjecture 2.9 as the manuscript cites it. Theorem 1.1 states that $O_p(G)=1$ implies nonzero augmented reduced rational homology of $\mathcal A_p(G)$ , for every finite group and prime, which implies non-contractibility (the Aschbacher-Smith rational form). The proof relies on published and preprint reductions (Aschbacher-Smith, Piterman-Smith, Piterman 2026 arXiv:2607.27500, Diaz Ramos 2026 arXiv:2303.15613v6) plus new frame-cycle constructions; it uses the classification of finite simple groups through those reductions. Not refereed here. No Lean formalization.

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

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Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Quillen 1978, Homotopy properties of the poset of nontrivial p-subgroups of a group

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Quillen 1978, Homotopy properties of the poset of nontrivial p-subgroups of a group evidence for this event

Quillen's conjecture on the poset of elementary abelian p-subgroups evidence for this event

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