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)
Manuscripts (2)
Nonsingular systems of equations over arbitrary groups
October 5, 2026 · source · cite as \cite{OAI:Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026}
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.
The Kervaire theorem for groups
September 24, 2026 · source · cite as \cite{OAI:The-Kervaire-Theorem-for-Groups-September-24-2026}
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.
Result 256 · 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.
Source: https://github.com/openai/math/blob/main/CONTENTS.md