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Math-2715 · mathematics · Group theory

An infinite residually finite group with finite presentation and 2-power torsion

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does an infinite group G exist with an ordinary finite presentation, residual finiteness and the property that every element has finite order equal to a power of2? No common exponent is prescribed.

Short English statement

Does an infinite group G exist with an ordinary finite presentation, residual finiteness and the property that every element has finite order equal to a power of2? No common exponent is prescribed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2715-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2715

The October main theorem asserts all four requested properties simultaneously: infinite, ordinarily finitely presented, residually finite and every element finite 2-power order. lean/docs/247.md covers the older generic periodic companion and explicitly does not document this stronger residual-finite 2-group main assertion. Within these assumptions, this is a claimed affirmative answer to the stated question.

An infinite residually finite group with finite presentation and 2-power torsion

OpenAI's claim

OpenAI presents a proof in the selected manuscript. 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.

Mathematical statement of the claimed result

Result 1.
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

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. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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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