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Math-2716 · mathematics · Group theory

A nontrivial group free-product Z cannot be normally generated by one element

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every nontrivial group A and every $w\in A*Z$, is $(A*Z)/\langle \langle w\rangle \rangle $ nontrivial?

Short English statement

For every nontrivial group A and every $w\in A*Z$, is $(A*Z)/\langle \langle w\rangle \rangle $ nontrivial?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2716-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2716

The Kervaire corollary states precisely nontriviality for every w when A is nontrivial. The main unimodular injectivity theorem handles degree plus/minus one; the other degrees retain a nontrivial cyclic quotient. lean/docs/256.md documents the selected unimodular injectivity theorem, not this all-word corollary as an exact formal statement. Within these assumptions, this is a claimed affirmative answer to the stated question.

A nontrivial group free-product Z cannot be normally generated by one element

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For every nontrivial group $G$, the free product $G*\mathbb Z$ is claimed not normally generated by a single element.

Mathematical statement of the claimed result

Result 1.
In the free product $H=A*\langle t\rangle$, a word $w$ is unimodular when its total exponent of $t$ equals $1$ or $-1$. For any coefficient group $A$ and such $w$, the source claims \[ A\longrightarrow H/\langle\!\langle w\rangle\!\rangle_H\text{ injective}. \] The Kervaire corollary drops the exponent assumption for a different conclusion: \[ A\ne1\quad\Longrightarrow\quad H/\langle\!\langle w\rangle\!\rangle_H\ne1\qquad(w\in H). \]

Outline of the argument

A traced spectral phase provides a unitary with a controlled large fixed space. Translating a putative coefficient relation into a planar surface yields one nontrivial boundary, which the phase calculation excludes; this prevents a normal generator of the required form.

Formal verification

For family 256, the selected statement is coefficient injectivity for a single relator of exponent sum $1$ or $-1$ in $A*\mathbb Z$. It underlies the Kervaire consequence but does not include the newer general nonsingular-system theorem. The listed Comparator files are Kervaire.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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