Quillen's conjecture on the poset of elementary abelian p-subgroups
Let be a finite group, a prime, and the poset of nontrivial elementary abelian -subgroups of . Quillen showed that a nontrivial normal -subgroup () makes contractible, and proved the converse for solvable groups, -rank at most two and groups of Lie type in defining characteristic. Aschbacher and Smith reduced the problem for to unitary-group cases and later work (Piterman, Smith, Diaz Ramos) narrowed it further, settling the rational form for odd primes, but the prime remained open. Conjecture: if is contractible, must ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Finite group theory; subgroup complexes and homotopy
- Posed by
- Daniel Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Advances in Mathematics 28 (1978), Conjecture 2.9
- Year posed
- 1978
- Years open
- 48y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 52 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every finite group and prime , if then (augmented, so the empty poset counts). This gives Quillen's conjecture and its stronger rational-homology form, and acyclicity over any field characterizes . 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 (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.
What the AI did
Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.
Verification
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 implies nonzero augmented reduced rational homology of , 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.