Result 247, Group theory

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Can a system governed by finitely many rules be infinite even though each of its symmetries repeats after finitely many steps? The manuscripts claim such an example, with additional structure that makes finite approximations informative.

What changes?

The group is finitely presented: finitely many generators and defining relations describe it. Yet it is infinite, and every element has order a power of two, with no uniform bound on those orders. The manuscript also reports residual finiteness, meaning any nonidentity element remains nonidentity in some finite quotient. This gives a negative answer to the finitely presented Burnside question even under this extra detectability requirement, not an example with uniformly bounded element orders.

What does that help mathematicians do?

The companion manuscript reports an infinite-dimensional, finitely presented associative algebra over the two-element field, without an identity. Every element has a power equal to zero, yet no fixed length makes all products vanish. It is also Jacobson radical. Adding an identity gives a finitely presented, infinite-dimensional algebra whose elements each satisfy a polynomial equation. These examples separate element-by-element constraints from global finiteness, ruling out the corresponding finite-presentation nilpotence and algebraic finiteness principles.

Are there practical applications?

The immediate value is foundational: finite descriptions and finite-order behavior need not guarantee finiteness, even when finite quotients distinguish elements. The reported residual-finiteness argument uses finite degree truncations of the associated graded algebra to detect group elements, including the central kernel. This connects the infinite construction to finite tests, without establishing an efficient computational method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

An infinite finitely presented residually finite 2-group

October 5, 2026 25 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

An infinite finitely presented periodic group

September 23, 2026 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. 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.

PDF LaTeX source

Cite (BibTeX)

@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's note on what the formalization covers, from lean/docs/247.md.

An infinite finitely presented residually finite 2-group and a finitely presented nil algebra

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

An infinite finitely presented periodic group

Scope

The finitely presented Burnside question asks whether a finitely presented group in which every element has finite order must be finite. 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.

Comparator links

Result	Comparator statement
Infinite finitely presented periodic group	PeriodicGroup.lean