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An infinite finitely presented residually finite 2-group
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Abstract
We prove that the infinite, ordinarily finitely presented periodic Steinberg group $\Gamma=\mathop{\mathrm{St}}\nolimits _{12}(R)$ of a companion paper is residually finite. Its finite-index subgroup $G=\ker(\Gamma\to\mathop{\mathrm{St}}\nolimits _{12}(\mathbb F_2))$ 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. Phases of long words in the companion's graded algebra allow finite degree truncations to detect all elements of Γ, including the central kernel.
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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, extending this paper's p = 2 result with a characteristic-p argument; its README notes that this paper also implies the residually finite periodic variant, and 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 the September companion's infinite finitely presented periodic Steinberg group; this paper's residual-finiteness and 2-group conclusions are not among the selected statements, and the nil- and radical-algebra conclusions are explicitly excluded. [family-level]
- OpenAI, PeriodicGroup.lean comparator statement (OpenAI Math Release, 2026-10-06). This comparator specifies both the rank-12 Steinberg witness and the abstract periodic-group existence claim, with
thm_mainleft assorryin this target file. [family-level] - 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 for an infinite finitely presented residually finite periodic group, and for a residually finite p-group for every prime, attributing both to Kitamura's repository; the formal proofs are linked externally and the reviewer did not rerun them. [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 statement, which contains this paper's main theorem as the case p = 2. [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 the existence of infinite finitely presented periodic groups as unsolved; Theorem 1.1 gives a residually finite example.
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
- Emergent Mind: family 247
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:An-infinite-finitely-presented-residually-finite-2-group-October-5-2026,
author = {{OpenAI}},
title = {{An infinite finitely presented residually finite 2-group}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf}{OAI:An-infinite-finitely-presented-residually-finite-2-group-October-5-2026}},
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 periodic group · September 23, 2026