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)

Manuscripts (2)

An infinite finitely presented residually finite 2-group

October 5, 2026 · source · cite as \cite{OAI:An-infinite-finitely-presented-residually-finite-2-group-October-5-2026}

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.

An infinite finitely presented periodic group

September 23, 2026 · source · cite as \cite{OAI:An-infinite-finitely-presented-periodic-group-September-23-2026}

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.

Result 247 · Group theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.