Browse › Group theory › 247 An infinite finitely presented residually finite 2-group and a finitely presented nil algebra
An infinite finitely presented periodic group
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Abstract
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. Its unitization is finitely presented, algebraic, and infinite-dimensional. These algebras answer the finitely presented nil- and radical-algebra nilpotence questions and the finite-presentation version of Kurosh's algebraic finiteness question negatively.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Kenta Kitamura, Infinite finitely presented residually finite p-groups for every prime (GitHub repository, October 2026). Claims an infinite finitely presented residually finite p-group for every prime p, building on this paper's construction; its Lean development reuses 63 unchanged modules from the openai/math repository, and the README states that ChatGPT, OpenAI Codex and Claude Code assisted. [AI-generated per source] [family-level]
Formalization
- OpenAI, Lean scope note for result family 247 (OpenAI Math Release, 2026-10-06). The scope note reports a Lean formalization of an infinite finitely presented periodic Steinberg group over a characteristic-two algebra, with no common exponent asserted, and explicitly excludes the separate nil- and radical-algebra conclusions.
- OpenAI, ComparatorChallenges/PeriodicGroup.lean (openai/math). Comparator statement: there is a
ZMod 2-algebra R such thatSteinberg 12 Ris infinite, finitely presented and periodic, and hence there is an infinite finitely presented periodic group; the proof is left assorryin this challenge file. - OpenAI, OAI/GroupTheory/PeriodicGroups/Main.lean (openai/math). Solution module that proves
thm_main(Theorem 1.1) from imported kernel-torsion and periodicity results. - Kenta Kitamura, Formalize the finitely presented Burnside problem and its variants (google-deepmind/formal-conjectures PR #7412, opened 2026-10-16, open). Adds Lean statements of Theorem 1.1 (citing this paper as [OAI26]) and two residually-finite variants, with
formal_prooflinks to an external solution; the reviewer approved but wrote that they did not rerun the verification. [family-level] - Kenta Kitamura, Add a conjecture: infinite finitely presented residually finite p-groups for every prime (google-deepmind/formal-conjectures issue #7284, 2026-10-13). Issue proposing the all-primes strengthening for formal-conjectures; it links the proof note and repository listed under follow-up works. [family-level]
Related prior literature mentioned by others
- Efim Zelmanov, Some open problems in the theory of infinite dimensional algebras (Journal of the Korean Mathematical Society 44(5), 1185–1195, September 2007; public text reproduction). Invoked by the AI Proved What? exposition; its Problems 1–3 concern finitely presented torsion groups and algebraic or nil algebras, and a subsequent paragraph considers whether residual finiteness might preclude such infinite torsion groups. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Unsolved Problems in Group Theory. The Kourovka Notebook, Problem 8.52 (A. Yu. Olshanskii) (arXiv:1401.0300v47, 2026-09-30). Lists "Do there exist infinite finitely presented periodic groups?" as unsolved; Theorem 1.1 addresses this question (Problem 6.3 of R. Bieri, on periodic groups of type FP∞, is also related).
- T. Adamczewski, B. Böhmler, R. Marczinzik, A counterexample to Köthe's conjecture and a question of Rowen (arXiv:2609.07996, 2026-09-07). A counterexample to Köthe's conjecture yields nil rings with non-nil matrix rings; this bears on why the paper proves nilpotence of every matrix in M_d(I) separately from nilpotence of elements of I.
Digests and expositions
- AI Proved What?, Finite rules, finite orders, still infinite? (Episode 247, October 2026). This AI-assisted video and transcript outline both papers' ring construction and Steinberg-kernel arguments, distinguish the full periodic group from its 2-group subgroup, and discuss the limited formalization scope. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 247
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:An-infinite-finitely-presented-periodic-group-September-23-2026,
author = {{OpenAI}},
title = {{An infinite finitely presented periodic group}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf}{OAI:An-infinite-finitely-presented-periodic-group-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/247.md. The Lean statements have not been checked by people against the paper's claims.
Family 247: An infinite finitely presented residually finite 2-group and a finitely presented nil algebra
Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽2-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.
Other papers in this family:
- An infinite finitely presented residually finite 2-group · October 5, 2026