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Math-2294 · mathematics · Group theory

The virtually-nilpotent/free-semigroup alternative

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Conjecture:A finitely presented group is either virtually nilpotent, or contains a free subsemigroup of order $2$.

Formulation references: [S1], [S2].

Short English statement

Conjecture:A finitely presented group is either virtually nilpotent, or contains a free subsemigroup of order $2$.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2294-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2294

An infinite finitely generated torsion group cannot be virtually nilpotent, since a finitely generated virtually nilpotent torsion group is finite. It cannot contain a free positive semigroup, since each proposed generator has finite order. Thus the main example refutes both alternatives. Within these assumptions, this is a claimed negative answer to the stated question.

The virtually-nilpotent/free-semigroup alternative

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: An infinite ordinarily finitely presented periodic group and an infinite-dimensional finitely presented nil algebra over $\mathbb F_2$ are claimed to exist; The periodic group of the companion is claimed residually finite, with an infinite finite-index finitely presented subgroup whose element orders are finite powers of two and unbounded. Together they give a claimed negative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
For a constructed unital associative algebra $R$ over $\mathbb F_2$, the Steinberg group $G=\operatorname{St}_{12}(R)$ is claimed to satisfy \[ |G|=\infty,\qquad G\text{ finitely presented},\qquad \forall g\in G\ \exists n\ge1:\ g^n=1. \] The companion corollary also assigns $G$ Kazhdan property $(T)$, hence nonamenability. Periodicity here permits element orders to vary.
Result 2.
The source claims existence of a group $G$ with an ordinary finite presentation such that \[ |G|=\infty,\qquad G\text{ residually finite},\qquad \forall g\in G\ \exists k\ge0:\ g^{2^k}=1. \] Thus all elements have finite order, and each order is a power of two; no common bounded exponent is asserted.

Outline of the argument

Finite local rules encode a hierarchy and a faithful computation tableau in admissible words. The resulting graded algebra has nil behavior without global nilpotence; a Steinberg-type group construction then transports these finite relations into an infinite group while forcing each element to have finite order.

Phases of long graded words make finite-degree truncations detect group elements. Faithful Steinberg strips, cyclic-shift comparisons, and long-entry commutator calculations treat the central kernel as well, so residual finiteness survives passage to the periodic two-group subgroup.

Formal verification

For family 247, the selected entry gives an infinite ordinarily finitely presented periodic group. A uniform exponent, the nil-algebra conclusions, and the later residual-finiteness strengthening are not part of it. The listed Comparator files are PeriodicGroup.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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