Result 256, Group theory

Nonsingular systems of equations over arbitrary groups

Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.

Lean formalization Proof

The bigger picture

Why it matters

Equations in group theory may require enlarging the group before they can be solved. The manuscript reports a condition guaranteeing that this enlargement can preserve every distinction between the original group elements.

What changes?

For any group and any finite system of equations using its elements as coefficients, the claimed guarantee is full row rank over the rational numbers of the exponent-sum matrix. This matrix records, for each equation and unknown, the total exponent of that unknown; full row rank means its rows are linearly independent. Under this condition, all equations have a simultaneous solution in an overgroup, a larger group containing an unchanged copy of the original.

What does that help mathematicians do?

A researcher imposing such equations can therefore rule out an unintended collapse of the coefficient group: two originally different elements do not become equal in the quotient defined by adjoining the unknowns and relations. This makes the rank test a certificate of consistency with the existing group structure, not a method for finding solutions inside the original group. The summary identifies this as resolving Howie's conjecture.

Are there practical applications?

Its immediate value is foundational: it clarifies when adding generators and relations can extend a group without destroying it. The companion manuscript reports that adding one generator and one relation cannot turn a nontrivial group into the trivial group, resolving Kervaire's conjecture. These are structural existence and non-collapse claims, rather than computational procedures for producing solutions.

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

Nonsingular systems of equations over arbitrary groups

October 5, 2026 16 pages

Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup. This proves Howie's conjecture on nonsingular systems.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026,
  author = {{OpenAI}},
  title = {{Nonsingular systems of equations over arbitrary groups}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf}{OAI:Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026}},
  year = {2026}
}

The Kervaire theorem for groups

September 24, 2026 13 pages

We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element. This resolves the Kervaire conjecture positively.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Kervaire-Theorem-for-Groups-September-24-2026,
  author = {{OpenAI}},
  title = {{The Kervaire theorem for groups}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf}{OAI:The-Kervaire-Theorem-for-Groups-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/256.md.

Nonsingular systems of equations over arbitrary groups

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

Scope

Kervaire's conjecture states that the free product of a nontrivial group with an infinite cyclic group cannot be normally generated by one element. The linked formalization proves the stronger coefficient-injectivity statement used in the paper: for any group AA and any relator in A∗ZA*\mathbb Z whose exponent sum in the cyclic generator is 11 or −1-1, the natural map from AA into the quotient by that relator is injective. This is the selected unimodular-relator result underlying the conjecture.

Comparator links

Result Comparator statement
Coefficient-group injectivity for unimodular one-relator quotients Kervaire.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.