---
id: "2026-10-06-openai-finitely-presented-burnside-periodic-group-claim"
url: "https://postcutoff.com/e/2026-10-06-openai-finitely-presented-burnside-periodic-group-claim/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Awaiting review]
verification: Peer-reviewed
sources: 9
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-finitely-presented-burnside-periodic-group-claim/

# Finitely presented Burnside problem answered negatively? (OpenAI, Lean)

Full title: OpenAI's math release claims an infinite finitely presented periodic group, answering the finitely presented Burnside problem negatively (a Steinberg group over an F₂-algebra, with property (T)); a residually finite 2-group version followed on 5 Oct; Lean covers the first

Family 247 of OpenAI's 6 Oct 2026 math release contains "An infinite finitely presented periodic group" (OpenAI, 23 Sept, 48 pages): "We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively." The group is St₁₂(R) for a unital associative F₂-algebra R; the paper adds that it has property (T) and is nonamenable, and that the same algebra gives a finitely presented nil algebra that is not nilpotent, answering the finite-presentation version of Kurosh's question. The companion "An infinite finitely presented residually finite 2-group" (5 Oct, 25 pages) proves the group residually finite and extracts a finite-index subgroup in which every element has 2-power order. Lean (challenge PeriodicGroup) covers the existence of an infinite finitely presented periodic group and was accepted in the oai-math-revised comparator run (25–26 Oct); the residual-finiteness and algebra results are not formalized.

## Key facts
- Manuscript 1 (abstract, verbatim): 'We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively. We also construct an infinite-dimensional finitely presented nonunital nil associative algebra over F₂ that is Jacobson radical but not nilpotent.' Theorem 1.1: 'the construction below gives a unital associative F₂-algebra R for which St₁₂(R) is infinite, finitely presented, and periodic.' Corollary 1.2: the group has Kazhdan's property (T), hence is nonamenable (via Ershov and Jaikin-Zapirain's theorem on Steinberg groups). 48 pages, dated 23 Sept 2026
- Manuscript 2 (5 Oct 2026, 25 pages): 'We prove that the infinite, ordinarily finitely presented periodic Steinberg group Γ = St₁₂(R) of a companion paper is residually finite. Its finite-index subgroup G = ker(Γ → St₁₂(F₂)) is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order. The orders of its elements are unbounded'
- Background as the paper recounts it: Burnside's questions started the study of finiteness under restrictions on element orders; Golod built infinite finitely generated periodic groups from nil algebras via the Golod–Shafarevich inequality; Novikov and Adian built infinite finitely generated groups of bounded exponent; Zelmanov solved the restricted Burnside problem; Ol'shanskii and Sapir 'explicitly distinguish the finitely presented periodic-group question'. Infinite finitely generated periodic groups with property (T) were known (Ershov); the new claim is the combination with an ordinary finite presentation
- Lean (OpenAI scope note lean/docs/247.md): 'The formalization gives a negative answer by constructing an infinite finitely presented periodic group, including a witness realized as a Steinberg group over an algebra of characteristic two. Periodicity means that each element has some finite order; no common exponent is asserted. The paper's separate nil-algebra and radical-algebra conclusions are outside these selected statements.' Challenge PeriodicGroup: 'accepted' in the oai-math-revised comparator run of 25–26 Oct. The residually finite 2-group is not in the scope note
- Follow-up: Kenta Kitamura's GitHub repository claims infinite finitely presented residually finite p-groups for every prime p, with a Lean proof; its README opens 'OpenAI solved the following finitely presented Burnside problem'. His formal-conjectures PR #7412 (opened 16 Oct, still open) would add the problem with 'research solved' tags, reusing OpenAI's theorem for the main statement
- Reactions: none from group theorists on this paper found (web, Hacker News, oai-math-revised notes) as of 29 Oct. In general reactions, Xiaolei Wu wrote that the release's group-theory section 'looked unreal. These were almost all the major open problems in my field!' (Proofs and Prompts, 8 Oct)

## What happened

Burnside asked in 1902 whether a finitely generated group in which every element has finite order must be finite. Golod and Shafarevich
showed in 1964 that it need not be, and Novikov and Adian (1968) found infinite examples of bounded exponent. Whether a group given by finitely many
generators and finitely many defining relations can be infinite and periodic remained open; Ol'shanskii and Sapir single it out as a
separate question.

OpenAI's 6 Oct release claims such a group: the Steinberg group St₁₂(R) over a specially built algebra R of characteristic two. The
same construction is said to give a finitely presented nil algebra that is not nilpotent. A second paper, dated 5 Oct, claims the group
is residually finite and contains a finite-index subgroup whose elements all have order a power of 2. The existence statement is
formalized in Lean, and a community comparator run accepted it. A volunteer has since claimed, with Lean, residually finite examples for
every prime.

## Why it matters

If the construction holds, one of the classical finiteness questions of group theory has a negative answer with an algebraic, explicit
witness. Because the Lean statement is short, specialists can check whether it says what group theorists mean: an ordinary finite
presentation and elements of finite order. That check, and any expert reading of the 48-page argument, has not been published. Our
coverage of the release had not listed this claim at all.

## What is disputed or not yet verified

- Verification: Lean formalization of the existence of an infinite finitely presented periodic group, accepted by an independent comparator run; residual finiteness and the algebra results are only in unrefereed manuscripts

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI (PDF): An infinite finitely presented periodic group (23 Sept 2026)](https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf) (github.com, paper)
2. [OpenAI (PDF): An infinite finitely presented residually finite 2-group (5 Oct 2026)](https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf) (github.com, paper)
3. [OpenAI Lean scope note for family 247](https://github.com/openai/math/blob/main/lean/docs/247.md) (github.com, code)
4. [oai-math-revised comparator results (PeriodicGroup accepted)](https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv) (github.com, code)
5. [GitHub: KitaKen1/all-prime-residually-finite-p-groups (Lean follow-up for every prime)](https://github.com/KitaKen1/all-prime-residually-finite-p-groups) (github.com, code)
6. [formal-conjectures PR #7412: Formalize the finitely presented Burnside problem and its variants](https://github.com/google-deepmind/formal-conjectures/pull/7412) (github.com, code)
7. [Wikipedia: Burnside problem (background)](https://en.wikipedia.org/wiki/Burnside_problem) (en.wikipedia.org, docs)
8. [Proofs and Prompts: 100+ reactions to 100+ solutions (8 Oct)](https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/) (proofsandprompts.com, discussion)
9. [oai-math-revised AGENT_NOTES for the periodic-group manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29 (filed): Created (CONTENTS entry, both PDFs (48 and 25 pp), Lean scope note, comparator results, Kitamura's repository and formal-conjectures PR, Proofs and Prompts reactions; web and HN searches found no specialist comment)

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