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The finitely presented Burnside problem: is every finitely presented periodic group finite?

A group is periodic if every element has finite order. Burnside (1902) asked how such conditions constrain finitely generated groups. Golod (1964) built infinite finitely generated periodic groups, Novikov and Adian infinite groups of bounded exponent, and Grigorchuk an infinite finitely generated 2-group, but none of these is finitely presented, and Ol'shanskii-Sapir's finitely presented torsion-by-cyclic groups have an infinite cyclic quotient. The finite-presentation version of Burnside's question asks whether a group given by finitely many generators and finitely many relators (in the ordinary sense, not within a variety) can be infinite while every element has finite order. Is every finitely presented periodic group finite?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Group theory; Burnside-type finiteness problems
Posed by
Recorded by Ol'shanskii and Sapir (Publ. IHES 2003, p. 45) and by Haettel and Osajda, in the line of Burnside's 1902 finiteness questions
Year posed
—
Years open
—
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
60 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims Theorem 1.1: there is a finitely presented graded F2\mathbb F_2-algebra RR such that St12(R)\mathrm{St}_{12}(R) is infinite, ordinarily finitely presented and periodic; Corollary 1.2 adds Kazhdan's property (T), so it is nonamenable. The companion claims St12(R)\mathrm{St}_{12}(R) is residually finite and that the kernel GG of the map to St12(F2)\mathrm{St}_{12}(\mathbb F_2) is an infinite finitely presented residually finite 2-group, necessarily of unbounded exponent. It does NOT give a finitely presented infinite group of bounded exponent (the bounded Burnside question for finite presentations), and element orders depend on the element.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The second (5 October 2026) builds on the first's algebra and local-rule hierarchy to add residual finiteness.

Verification

No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against the question: an explicit unital F2\mathbb F_2-algebra RR for which the Steinberg group St12(R)\mathrm{St}_{12}(R) is infinite, ordinarily finitely presented and periodic. The Lean challenge ComparatorChallenges/PeriodicGroup.json (theorem OAI.SourceBurnside.thm_main, solution module OAI.GroupTheory.PeriodicGroups.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/247.md. Its statement was read here: a ring RR that is an F2\mathbb F_2-algebra with the Steinberg group on 12 indices (presented by the addition and commutator relations) infinite, finitely presented and periodic, together with the bare statement that some infinite finitely presented periodic group exists. That is the headline claim; no uniform exponent is asserted. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The residual-finiteness companion and the nil-algebra conclusions are not formalised.

Sources

Changelog1 change

Discussion