Nonsingular systems of equations over arbitrary groups

Claims every nonsingular system of equations over any group is solvable in a larger group, settling Howie's conjecture.

2 papers September 24, 2026 Explained by AI

An equation over a group mixes known group elements with unknowns. It is solvable if a larger group contains the original one and values for the unknowns. The first paper claims this always works when a simple count of the unknowns, done with ordinary integers, gives independent rows. A companion paper proves the one-equation case behind Kervaire's conjecture, with a Lean formalization listed.

First step

1962

Murray Gerstenhaber and Oscar Rothaus, who solved nonsingular systems over finite groups and compact Lie groups

Significance

Major

Kervaire's conjecture and Howie's conjecture are long-standing open problems in combinatorial group theory.

Trust

Formal proof listed

Field

Group theory

2 papers

The explanation

Take a group A with an element a ≠ 1. Add a new unknown t and impose the equation t a t⁻¹ = 1. Conjugating back gives a = 1. So this equation destroys a, and no larger group can solve it while keeping a. Count how often t appears, with signs: +1 − 1 = 0. That count is the exponent sum. Now try t a t a² t⁻¹ = 1. Its exponent sum is 1 + 1 − 1 = 1. The Kervaire–Laudenbach question asks whether a nonzero exponent sum always prevents such collapse.

An equation over a group G is a word mixing elements of G with unknownsSource 1. A solution is a larger group containing a copy of G, with values for the unknowns that make every word equal 1Source 1. For several equations, the exponent sums form an integer matrix. For example, x₁x₂g and x₁x₂⁻¹h give rows (1, 1) and (1, −1). The determinant is 1·(−1) − 1·1 = −2, which is not zero. So the rows are independent. A system is nonsingular when its rows have full rank over the rationalsSource 2.

Gerstenhaber and Rothaus in 1962 solved nonsingular systems over finite groups and compact Lie groupsSource 3. Howie solved them over locally indicable groupsSource 4. Klyachko treated one equation over torsion-free groups with exponent sum ±1Source 5. The obstacle was arbitrary coefficient groups. These may have torsion, be uncountable, or have no nice finite models.

The companion paper proves the one-variable case: for every group A and every word with exponent sum ±1, A embeds in the quotientSource 6. The Kervaire conjecture follows: one new generator and one new relation never kill a nontrivial groupSource 7. For other exponent sums d, the quotient maps onto the nontrivial cyclic group of order |d|, or onto the integers when d = 0Source 8.

The first paper claims the general statement. If the exponent-sum matrix has full row rank, G embeds in the quotientSource 9, so the system has a solution in an overgroupSource 10. This is Howie's conjectureSource 11. The proof turns a hypothetical lost element into a flat surface with disks and bands. Full row rank forces matching numbers of positive and negative disks for each equationSource 12. A spectral phase function, measured through the regular representation of G, then forces the lost element to be the identity.

How sure is this? A computer proof checker called Lean has a formal proof of the one-relator theorem with exponent sum ±1. The many-equation theorem has no Lean formalization. No specialist review of either paper has been published.

What it doesn't show. This does not give a Lean proof of the many-equation theorem, or say the larger group is finite or specially structured.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. Lean covers the unimodular one-relator theorem behind Kervaire's conjecture, not the many-equation theorem. No specialist review is published.

Words used

Equation over a group: A word mixing fixed group elements with unknowns, to be set equal to the identity.
Overgroup: A larger group that contains a copy of the original group.
Exponent sum: The total number of times an unknown appears, counting its inverse as minus one.
Nonsingular system: A system whose matrix of exponent sums has independent rows over the rational numbers.
Normal closure: The smallest normal subgroup containing given elements; quotienting by it imposes those elements as relations.
Torsion: Elements other than the identity that have finite order.
Locally indicable: Every nontrivial finitely generated subgroup maps onto the integers.

The original papers (2)

Nonsingular systems of equations over arbitrary groups (PDF, opens in a new tab)
October 5, 2026
The Kervaire theorem for groups (PDF, opens in a new tab)
September 24, 2026

Lean notes

All quoted lines (12)

An equation over a group permits coefficients from that group and asks whether the unknowns can be realized in a larger group without identifying distinct coefficients.
Nonsingular systems of equations over arbitrary groups Back to text
The system is nonsingular if $\operatorname{rank}_{\mathbb Q}A=m$ .
Nonsingular systems of equations over arbitrary groups Back to text
The foundational system theorem is due to Gerstenhaber and Rothaus.
Nonsingular systems of equations over arbitrary groups Back to text
Howie proved solvability of finite independent systems over locally indicable groups; locally indicable means that every nontrivial finitely generated subgroup maps onto $\mathbb Z$ .
Nonsingular systems of equations over arbitrary groups Back to text
Klyachko's theorem treats a single equation in one variable over a torsion-free group when the exponent sum is $\pm1$ .
Nonsingular systems of equations over arbitrary groups Back to text
For every group $A$ and every unimodular word $w\in A*\langle t\rangle$ , the canonical homomorphism $A\longrightarrow (A*\langle t\rangle)/\langle\!\langle w\rangle\!\rangle_H$ is injective.
The Kervaire theorem for groups Back to text
We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element.
The Kervaire theorem for groups Back to text
For every other integer $d=p(w)$ , the map $p$ induces a surjection from the quotient onto the nontrivial group $\Z/d\Z$ , where $\Z/0\Z=\Z$ .
The Kervaire theorem for groups Back to text
If their exponent-sum matrix has row rank $m$ over $\mathbb Q$ , then the canonical homomorphism $G\longrightarrow (G*F_n)/\langle\!\langle w_1,\ldots,w_m\rangle\!\rangle$ is injective.
Nonsingular systems of equations over arbitrary groups Back to text
Equivalently, the system has a simultaneous solution in an overgroup of $G$ .
Nonsingular systems of equations over arbitrary groups Back to text
The general solvability assertion for nonsingular systems is commonly called Howie's conjecture; see for this formulation and terminology.
Nonsingular systems of equations over arbitrary groups Back to text
First, full row rank forces equal numbers of positive and negative disks of each relator type.
Nonsingular systems of equations over arbitrary groups Back to text

Other sources

openai/math, Lean scope note for family 256 for any group $A$ and any relator in $A*\mathbb Z$ whose exponent sum in the cyclic generator is $1$ or $-1$ , the natural map from $A$ into the quotient by that relator is injective.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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