AI is changing more than mathematicians’ tools. It is changing the profession itself.
AI is already helping to build proofs and solve research problems. Our starting point is that the profession will not stay as it is today. The mathematical community will have to adapt — in research, education, and the way it assesses research.
MAIDENHEAD PROJECTSolutions, methods, and mathematical explanations.
01 / HOW WE WORK
Trying the roles ourselves
Our current focus is algebraic group theory. We welcome collaborators from other areas of mathematics and other sciences.
We see five parts of this changing profession: the philosopher of a field, the connector, the research architect, the explainer, and the mentor.
Our task is to understand how to combine these roles while preserving deep, independent thinking, developing links between mathematics, other sciences, and practical problems, and making mathematics easier for people to understand. We also need to learn how to prepare the next generation for this work.
To understand how these roles would work, we need to try each of them ourselves. Here is what we are doing so far. The five roles in our manifesto ↗
i.
The philosopher of a field
Depth
Understands a field’s main ideas, methods, and structure, keeps up with what is happening across it, and actively develops and changes it. They read the field’s “newspaper” every day while thinking about future directions, asking what we really understand and where we should go next.
What we do
The Maidenhead Project itself is our attempt to take on this role. We follow developments in mathematics and AI and think about where they could lead. Our research and our manifesto explore what these developments mean for the questions we ask, how we work, and how we train new mathematicians.
ii.
The connector
Breadth and possible applications
Finds meaningful links between areas of mathematics, other sciences, and practical problems, and actively develops them, changing the fields involved.
An invitation
This is a role we still need to develop. Our own focus is algebraic group theory. We invite people from other areas of mathematics and from other sciences to join us, bring their questions and perspectives, and build connections together.
Plans the research process, divides tasks between people and AI, and organises verification. Responsibility cannot be passed to a tool along with the task.
What we do
We are building a system of AI agents for mathematical research and testing it on open problems from the Kourovka Notebook in group theory. Some of our results come from this system and some from working with a model directly in its web interface; each record says which. We plan to publish the code so other mathematicians can use it, examine how it works, and improve it.
Often the model does the proving: for 20 of our 24 solved problems, it worked on its own after the first prompt. Our part is to choose the problems, organise verification, state exactly what each result covers, and take responsibility for what we publish.
Every record shows its verification status and scope, the model and setup used, and how much human guidance it needed. See the results ↓
On harder problems, agents and people work from one shared record: every note states its status, its scope and what it does not claim, nobody closes their own line of work, and a person sets the direction and makes the go/no-go calls. Alexei Miasnikov has also used the setup on his own research, with three new results on the Diophantine problem in wreath products (unpublished; link TBA).
A recent paper
In October, Elena Bunina, Winston Lee, Alexei Miasnikov and Kyrylo Muliarchyk posted Failure of direct finiteness for group algebras in every odd characteristic (arXiv:2610.22536). ChatGPT was used extensively for exploration, checking, literature search and drafting. The paper builds on three AI-generated manuscripts, cites each at a fixed revision, and states that the named authors are responsible for every statement, proof and citation. See it in the results ↓
CODE, PROOFS & RESEARCH NOTES
Our repository is currently private. We’re preparing it for public release.
GitHub · Coming soon
iv.
The explainer
Understanding and openness
Identifies the main idea, the real difficulty, and a useful example, and can discuss the same mathematics with specialists, students, other scientists, and people without specialist training.
What we do
54 of our 56 problem records include a short summary of what was established and its central idea, and 34 have a full write-up you can read on this site.
Seminar and paper club
At Bar-Ilan University, Elena Bunina leads a weekly seminar on algebras, whose discussions include what AI has achieved in mathematics.
In London, Stan Fedotov organises Nebius Academy’s AI-Assisted Math Research Paper Club. Free and open to anyone interested, the group discusses recent papers and news, and examines what AI contributes to mathematical discovery.
Helps others become independent researchers and develop mathematical taste, and passes on an approach to work: honesty, persistence, and responsibility.
Our commitment
All professors in our group already advise many students, including PhD students, and are committed to continuing this work. Training the next generation of mathematicians is a core principle of the Maidenhead Project.
Courses
Elena Bunina teaches an “AI in Math” elective at Bar-Ilan University, and Alexei Miasnikov teaches one at Stevens Institute of Technology. Open course materials ↗
Stan Fedotov teaches on Nebius Academy’s AI-Assisted Mathematical Discovery course at the London Institute for Mathematical Sciences. Participants use coding agents and machine learning to run mathematical experiments and search for examples and counterexamples.
02 / RESULTS
Problems and solutions
Our first problem source is the Kourovka Notebook. Results from other collections will be added here.
Many problems on lists of open problems are now within reach of current AI models. We run our agents over such lists, starting with the Kourovka Notebook, check every answer, and publish it here with its exact scope. We do not post these answers to arXiv as new papers; finding the right way to bring them into the wider field is part of the work. When a solution leads to interesting mathematics, we write that up as usual. What remains is a “purified” collection, in Alexei Miasnikov’s phrase: the problems that resist this, which is where sustained human work goes.
Solving also tests the lists themselves. For 24 problems listed as open, our setup found that an answer had already been published; those records credit the earlier authors and are not counted as ours.
Our manifesto asks three questions of every new proof: what has been established and how its correctness was checked, what the main idea is, and what it helps us understand. Our records answer the first as precisely as we can: verification status, exact scope, a summary, and the full write-up where available. The other two — why a result matters and what it helps us understand — are still thin in most records.
Selected results
arXiv:2610.22536 · math.GR · 15 October 2026Preprint
Failure of direct finiteness for group algebras in every odd characteristic
Elena I. Bunina, Winston Lee, Alexei Miasnikov, Kyrylo Muliarchyk
For every odd prime p, a finitely presented group whose group algebra over 𝔽ₚ has elements with ab = 1 but ba ≠ 1. Kaplansky asked in 1969 whether a one-sided inverse in such an algebra must be two-sided. Together with OpenAI’s preprint for characteristic 2, this answers the question negatively in every positive characteristic; neither paper has been refereed yet. The groups are not sofic; torsion-freeness is not claimed.
ChatGPT · exploration, checking, literature search and drafting · the named authors are responsible for every statement and proof
A non-powerful subgroup of p-th powers for every odd prime
The editors accepted the case p = 3. Kyrylo Muliarchyk then asked the model to extend the construction to every odd prime by analogy, and pressed it when it resisted. The uniform version is team-checked and with the editors.
A solvable group with a transitive action in which every element taking one given point to another fixes exactly one point. This refutes Müller’s conjecture already for solvable groups; the fixed-point calculation is carried out directly on twelve affine blocks.
Our agents found counterexamples for every d ≥ 2, not only d = 2, built from finite powers of PSL₂(11). The write-up also shows that the proposed bound does hold at d = 1, so the stated scope is exact.
Our setup proved this sharp bound; checking the literature then turned up a note, deposited on Zenodo on 2 August 2026, with the same rank argument. The problem was no longer open, and the record credits the earlier author.
We will also add Alexei Miasnikov’s three results on the Diophantine problem in wreath products, obtained using our research setup. The work is unpublished; link TBA.
All records
Explore the records by source, verification status, or model. Open a solution summary to read the argument and its available proof.
Problem source
Verification
Showing 56 of 56 problems
Problems, sources, verification status, and solving setup
Problem 21.137 — A non-powerful subgroup of p-th powers for every odd prime
For each odd prime p, an explicit group of order p³ᵖ⁺¹ and exponent p² has its set of p-th powers equal to a subgroup isomorphic to UT₃(F_p) × C_pᵖ⁻¹, which is nonabelian of exponent p. This gives a uniform negative answer to the odd-prime part of the problem. The Kourovka editors have accepted the case p = 3; the uniform write-up for all odd primes is team-checked and has been sent to the editors.
Review status: Editor-accepted. Model and setup attribution supplied by the project team.
Human contribution: Kyrylo Muliarchyk asked the model to extend the construction to all odd primes by analogy, and pressed it to pursue the generalization after initial resistance.
Problem 19.20 — Counting endomorphisms and partial isomorphisms
A nonabelian group of order 512 has equally many endomorphisms and partial isomorphisms, giving a negative answer. The endomorphism count has a direct proof; the partial-isomorphism count uses exhaustive finite quadratic-map enumeration. The PDF includes a counting certificate and describes its verification; no claim of minimal order is made.
Review status: Editor-accepted. Listed as a solution in the Kourovka editors’ repository, checked on 18 September 2026. Kyrylo Muliarchyk confirms that this is the project’s result. The local PDF may retain an earlier review label.
Human contribution: Kyrylo Muliarchyk requested counterexamples and recalls asking substantive follow-up questions, beyond continuation requests; the exact questions are not recorded.
Problem 21.148 — Right-relatively convex subgroups with a distributive lattice
In the group of increasing bijections of the rationals with bounded displacement, the right-relatively convex subgroups are exactly the isolated subgroups and exactly the convex ℓ-subgroups. Their lattice is distributive but not a chain, giving a negative answer. The main ingredient is an orbital sandwich theorem, proved with a coding construction adapted from Hyde, Jonušas, Mitchell and Péresse. The Kourovka editors now list Kyrylo Muliarchyk’s paper as a solution.
Review status: Editor-accepted. Model and setup attribution supplied by the project team.
Human contribution: Kyrylo Muliarchyk requested counterexamples and recalls asking substantive follow-up questions, beyond continuation requests; the exact questions are not recorded.
Problem 8.74 — Intersections of good subnormal subgroups
The submitted argument constructs two good subnormal subgroups whose intersection is not good, in a countable locally finite 2-group of derived length at most three. It proves goodness using an explicit augmentation-ideal bound.
Review status: Editor-accepted. Listed as a solution in the Kourovka editors’ repository, checked on 18 September 2026. Kyrylo Muliarchyk confirms that this is the project’s result. The local PDF may retain an earlier review label.
Problem 16.53 — Generating a group from two coprime subgroups
For every d ≥ 2, there are finite groups G = ⟨A, B⟩ with coprime subgroup orders, d(A) = d(B) = d, and d(G) = d + 2. Thus the counterexamples are not limited to d = 2. The construction uses finite powers of PSL₂(11), with A and B metabelian of exponents 6 and 55. At d = 1 the proposed bound holds: two cyclic subgroups together provide at most two generators.
Review status: Editor-accepted. Scope follows Theorem 1 of the editor-hosted paper: every integer d ≥ 2. Local PDF updated to that version on 20 September 2026.
Problem 10.46 — Double commutators with transvections over a noncommutative ring
For every field k, including characteristic zero, and every n ≥ 2, a finitely presented noncommutative k-algebra admits an invertible matrix whose nonempty iterated commutators with nonidentity unipotent matrices are nonunipotent and have infinite order. In particular, this refutes the proposed double-commutator assertion. The argument uses Gerasimov’s theorem and a free-product tree; it does not assert a counterexample over commutative rings.
Review status: Editor-accepted. The editor-hosted version covers all field characteristics. Local PDF updated to that version on 20 September 2026.
Problem 20.75 — Abelian normal subgroups with few generators
One explicit finite 2-group is a counterexample to both Problems 20.75 and 20.76. For e = 256,363,762, the group has order 2^(144e + 1); every abelian normal subgroup has order at most 2^(42e) and at most 42e generators, while an elementary abelian subgroup has order 2^(85e) and requires 85e generators. This simultaneously refutes the generator bound and the square-order bound. The paper also proves existence of a smaller counterexample of order 2^145.
Review status: Editor-accepted. Both records refer to the same counterexample and joint paper, not two independent constructions. The model and setup retain the previously recorded investigation for this problem. Local PDF updated to the joint editor-hosted paper on 20 September 2026.
Problem 21.99 — Transporters with exactly one fixed point
A solvable group of order 46,875,000,000 has a faithful transitive action of degree 23,437,500 with distinct points α and β such that every element taking α to β fixes exactly one point. This refutes Müller’s conjecture already for solvable groups. The group is an extension of F₅¹² by a group of order 192, and the fixed-point calculation is carried out directly on twelve affine blocks.
Review status: Editor-accepted. Listed as a solution in the Kourovka editors’ repository, checked on 18 September 2026. Kyrylo Muliarchyk confirms that this is the project’s result. The local PDF may retain an earlier review label.
Human contribution: Kyrylo Muliarchyk requested counterexamples in the web interface. Classified as Five-prompts by the project team.
Problem 11.23 — e-automorphisms and nil-automorphisms
The order-four cyclic shift on the fourth direct power of the first Grigorchuk group gives an e-automorphism that is not a nil-automorphism. The write-up includes a finite section certificate for non-nilness.
Review status: Editor-accepted. Listed as a solution in the Kourovka editors’ repository, checked on 18 September 2026. Kyrylo Muliarchyk confirms that this is the project’s result. The local PDF may retain an earlier review label.
Problem 20.30 — Group order and the largest conjugacy class
An explicit perfect centreless finite group has order greater than the square of its largest conjugacy-class size: the ratio is 250/3. The construction uses a finite Lie algebra over 𝔽₅.
Review status: Editor-accepted. Listed as a solution in the Kourovka editors’ repository, checked on 18 September 2026. Kyrylo Muliarchyk confirms that this is the project’s result. The local PDF may retain an earlier review label.
Problem 20.76 — Bounds on abelian subgroups of finite p-groups
One explicit finite 2-group is a counterexample to both Problems 20.75 and 20.76. For e = 256,363,762, the group has order 2^(144e + 1); every abelian normal subgroup has order at most 2^(42e) and at most 42e generators, while an elementary abelian subgroup has order 2^(85e) and requires 85e generators. This simultaneously refutes the generator bound and the square-order bound. The paper also proves existence of a smaller counterexample of order 2^145.
Review status: Editor-accepted. Both records refer to the same counterexample and joint paper, not two independent constructions. The model and setup retain the previously recorded investigation for this problem. Local PDF updated to the joint editor-hosted paper on 20 September 2026.
Problem 14.67 — Largest centralizers and nilpotent subgroups
A counterexample of order 660,602,880 gives a negative answer, even when the nilpotent subgroup is normal and has nilpotency class two. An involution has a largest centralizer among nonidentity elements but does not centralize this subgroup. The write-up gives a self-contained proof without computer enumeration.
Review status: Team-checked. Model and setup attribution supplied by the project team.
Problem 21.115 — A sharp bound for complements of coset unions
If n left or right cosets do not cover a finite group G, their complement has at least |G|/2ⁿ elements. The proof establishes this sharp bound through matrix-rank estimates, with equality attained by coordinate index-two cosets. A note by another author deposited on Zenodo on 2 August 2026 gives the same rank argument, so the problem was solved before our write-up.
Review status: Solved elsewhere. Model and setup attribution supplied by the project team.
Kyrylo Muliarchyk identifies this problem as already solved in Jens Harlander’s Embeddings into efficient groups. The project’s later write-up is retained as a reference, not counted as a new solution. The write-up describes N/[F, N] for a minimal free presentation of a finitely presented group. Its abstract isomorphism type is determined by the Schur multiplier and a free abelian summand, independently of the chosen minimal presentation. It also discusses the limits of computing these invariants from an arbitrary finite presentation.
Review status: Solved elsewhere. Reclassified as solved elsewhere following Kyrylo Muliarchyk’s literature correction. Model and setup refer to our later write-up, not the published solution.
Problem 14.23 — Bounds for bases of fixed subgroups
The answer depends on whether the rank is fixed. For each fixed rank, a computable bound on basis lengths follows from existing algorithms for fixed subgroups. A single bound for all ranks is impossible: an explicit family of automorphisms of norm 2 provides counterexamples. The Kourovka editors have since accepted an independent solution by Olga Kharlampovich; our write-up remains available for reference.
Review status: Solved elsewhere. Model and setup attribution supplied by the project team.
Problem 14.15 — Commutator width of automorphism groups
A counterexample is already available from the literature, as confirmed by Kyrylo Muliarchyk. Our later write-up is retained for reference, not counted as a new project solution. The commutator width of Aut(Fₙ) is infinite for every n ≥ 3. The argument draws on published results about quasimorphisms and verbal width, connecting the Notebook question to existing theorems rather than claiming a new general theorem.
Review status: Solved elsewhere. Reclassified as solved elsewhere following Kyrylo Muliarchyk’s correction. The linked paper is supporting literature cited in our write-up; model and setup describe our later exposition, not authorship of the published result.
Problem 21.68 — Semiabelian groups and monomial representations
Aluna Rizzoli’s September 2026 paper gives a semiabelian nonmonomial group of order 2,592. The Kourovka editors list it as a solution. The earlier GPT-6 Astra web investigation is retained as provenance, not a claim of priority for the published result.
Review status: Solved elsewhere. Model and setup attribution supplied by the project team.
Problem 6.38 — Subgroups meeting every conjugacy class of GL₂
The invertible monomial subgroup of GL₂ over the algebraic closure of F₂ meets every conjugacy class but is not parabolic. This refutes the general-linear assertion of part (b) as literally stated, in dimension two and characteristic two. The same monomial family appears in the Notebook archive for part (a), so no novelty is claimed, and other groups of Lie type are not addressed.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 17.32 — A Cayley–Hamilton analogue for free groups
Olga Kharlampovich is credited with the solution, as clarified by Kyrylo Muliarchyk. The editors’ repository links a rank-two counterexample. Our separate web investigation and later local write-up are retained as project history, not counted as a new solution. The local write-up discusses examples at every rank n ≥ 2 and related variants.
Review status: Solved elsewhere. Attribution corrected on 20 September 2026 following Kyrylo Muliarchyk’s clarification. He requested counterexamples using GPT-5.6 Pro in the web interface. The local PDF is the separate legacy-agent write-up previously recorded with GPT-5.6 Sol.
Human contribution: Kyrylo Muliarchyk requested counterexamples in the web interface.
Problem 9.4 — Groups generating classes of quasigroups
Negative answers to all three cases: varieties, quasivarieties, and pseudovarieties. The last case uses the original author’s convention of classes defined by disjunctive identities, rather than finite HSP closure.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 9.45 — Orthogonal bases of rational lattices
An explicit, finite arithmetic criterion decides whether the lattice ℤⁿ + ℤa has an orthogonal integer basis, for every rational vector a. When the criterion holds, it constructs a basis. No polynomial-time bound or structural classification is claimed.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
For every prime p, finite p-groups of unbounded nilpotency class generate mixed varieties that are not ordinary varieties of groups. The obstruction is failure of closure under subgroups.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 10.43 — Adjoint homomorphisms in even degree
Adjoint representations give explicit homomorphisms violating the stated condition for every even degree n ≥ 4 over a nonzero commutative ring in which 2 is invertible. This addresses the permitted particular cases, not the full representation classification.
Review status: Partial progress. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 11.99 — Counting characters through group equations
A criterion using only the group order and counts of solutions to commutator equations determines whether a finite group has an irreducible character of defect zero for more than one prime. It also counts those characters, without taking a character table as input.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 13.1 — Coefficient reduction in integral group rings
For every finite group G and prime p, terminating algorithms compute the image of U(ℤG) → U(𝔽ₚG), representatives of its kernel cosets, and a finite generating set for the kernel. This is an algorithmic answer; adequacy to a stronger intended structural classification and historical priority remain open.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 18.109 — Property (T) and finite Lie-rank-one quotients
A group with Kazhdan’s property (T) surjects onto PSU₃(q) for every prime q congruent to 7 modulo 12, giving infinitely many nonisomorphic finite simple quotients. Rank is understood in the relative, or BN-pair, sense.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 18.59 — A linear obstruction to a periodic group construction
A linear-algebra obstruction over 𝔽₂ rules out the group required by the problem. The negative answer does not require the ambient group to be periodic.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 19.3 — Near Frattini subgroups of amalgamated products
For proper amalgams, parts (a), (b), (c), (e), and (f) are affirmative, while (d) has a counterexample. Allowing collapsed amalgams changes several answers. The printed hypotheses of (g) are inconsistent; no intended repair of (g) is declared solved.
Review status: Partial progress. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 21.132 — Infinite Golod groups with finite centre
For every prime p, an infinite, three-generated, residually finite Golod p-group is constructed with centre of order p. The construction starts from Golod’s nil-algebra method.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 18.120 — Factorizations of finite p-groups
An unverified draft proposes a group of order 2³² with an exact factorization P = AB, where A is abelian and B has class two, but the required abelian subgroup B₀ cannot exist. The proposed negative answer includes A ∩ B = 1.
Review status: Pending review. Unverified draft from the September 17 campaign. Earlier automated review reports do not override the project’s current verification status.
An unverified draft proposes a counterexample of order 719,323,136 at p = 7 to the inequality ℓᵖ ≥ kᵖ⁻¹. The argument computes the Sylow number and the number of 7-elements explicitly.
Review status: Pending review. Unverified draft from the September 17 campaign. Earlier automated review reports do not override the project’s current verification status.
Problem 20.122 — Minimal triple intersections of nilpotent subgroups
An unverified draft proposes a soluble group of order 96 refuting the inclusion-minimal assertions in (b) and (c). Part (a) and the minimum-order assertion in (c) remain open.
Review status: Pending review. Unverified draft from the September 17 campaign. Earlier automated review reports do not override the project’s current verification status.
Problem 21.113 — A class function that is not a character
An unverified draft proposes a group of order 58,320 for which the class function in part (a), at p = 2, has inner product −1 with an irreducible character. The conditional projectivity question in (b) is not settled.
Review status: Pending review. Unverified draft from the September 17 campaign. Earlier automated review reports do not override the project’s current verification status.
Subgroups H normal in Q inside four copies of UT₃(𝔽₂) both project onto every factor, yet Q/H is a nonregular 2-group of order 32. This gives a negative answer.
Review status: Team-checked. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 20.4 — Normal p-subgroups of soluble cut groups
The upper triangular matrix group B₆(𝔽₅) is a soluble cut group whose largest normal 5-subgroup has exponent 25. This refutes part (c) at p = 5; no answer to (a) or (b) is asserted.
Review status: Partial progress. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
Problem 20.71 — Vertex orbits and graph reconstruction cards
A connected graph on nine vertices has exactly four isomorphism types of vertex-deleted cards but five automorphism orbits. This answers the k = 4 case; no answer for k = 2 or 3 is claimed.
Review status: Partial progress. Verified at the stated scope in the project’s 18 September 2026 register. The PDF may retain an earlier candidate or review-pending label; editor acceptance and priority are separate questions.
An explicit soluble group has a nonsoluble regular subgroup in its holomorph. The coordinate construction comes from Di Matteo, Ferrara and Trombetti’s July 2026 paper; the project supplies a direct-action certificate and makes no priority claim.
Review status: Solved elsewhere. Model and setup attribution supplied by the project team.
Bludov, Kopytov and Rhemtulla (2009), Example 3.1, give the negative answer. The screening checked its fit to this exact Notebook question; no new project proof is claimed.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 6.5 — Constructibility and homological finiteness
Martínez-Pérez and Nucinkis (2010), Theorem 1.1, supply the constructibility theorem covering soluble groups of integral type FP∞. The project identified the existing result.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Montinaro (2007), Theorem 2.4(1), gives the classification: Desarguesian or Lüneburg translation planes. The screening checked the theorem’s scope; no new proof or editor recognition is claimed.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 12.17 — Autostable periodic abelian groups
Melnikov and Ng (2018), Theorems 1.3–1.4, give an exact effective description of computably categorical torsion abelian groups. This is an effective criterion, not an algebraic normal-form list.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Jha and Johnson (1998), Theorem 1.2, classify the planes as Desarguesian, Hall of order 9, or Hering of order 27. The screening checked the full case scope without independently auditing the proof.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 13.12 — Automorphism groups of hyperbolic groups
Carette (2011), Theorem 6.3, establishes finite presentability in the full hyperbolic-group scope. The project identified the published answer; no fresh proof audit is claimed.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Tent (2016), Corollary 2.3, gives a negative answer using sharply three-transitive groups and the fixed-point behaviour of involutions. The screening checked the exact scope.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Shakhova (2018) gives a finite group whose Levi class has infinite axiomatic rank, answering the question negatively. The publisher’s abstract was authenticated; the full proof was not available for review.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 16.56 — Groups with restricted element orders
Jabara, Lytkina, Mamontov and Mazurov (2014) give the affirmative answer. The published statement was checked against the required spectrum; the project did not independently inspect the full proof.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 17.75 — Actions of the Monster on finite 3-groups
Lee and Popiel (2023), Section 3, give a negative answer: elements of order 41 obstruct the proposed action. The September editor update recognizes the answer; the project did not rerun the computational inputs.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Schmid (2018) rules out the requested noncyclic irreducible 5′-subgroup. The screening authenticated the published full-answer claim, without reconstructing all proof dependencies.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 18.44 — Faithful irreducible representations
Jones and Keller (2020) give the affirmative answer. The project authenticated the publisher’s exact statement, without an independent proof audit.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 19.4 — Near Frattini subgroups of free-group amalgams
Allenby (2000), Theorem 4, gives a family of cyclic amalgams answering both parts negatively, with the printed allowance of rank-one free factors. This is an existing construction, not a new project result.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 19.5 — Near Frattini subgroups of abelian-group amalgams
Allenby (2000), Theorem 4, gives a negative answer using a cyclic amalgam with a nontrivial near Frattini subgroup. The same published family also answers 19.4.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 19.41 — Conjugacy separability of free-by-cyclic groups
Dahmani, Hughes, Kudlińska and Touikan prove that finitely generated free-by-cyclic groups are conjugacy separable. The full scope appears in version 2 and is recognized by the September editor update. The project did not independently audit the proof.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 19.94 — Distributions of elements in free metabelian groups
Ionin and Semidetnov give a rank-two counterexample: identical distributions on finite metabelian groups do not force the same automorphism orbit. The September editor update records the negative answer.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Problem 19.96 — Definability of bases and primitive elements
Kharlampovich and Miasnikov (2025), Theorem 3.21, define ordered free bases without parameters in every finite rank at least two. Projection then defines primitive elements, answering both parts.
Review status: Solved elsewhere. The project identified an existing answer and checked its relevance at the scope recorded in the summary. This does not count as a new project solution.
Human contribution: Original proof credited to the authors in the summary.
Updated 20 September 2026. Verification follows the project’s register at each result’s stated scope.
Contribution classes are qualitative, not literal prompt counts or measures of proof verification. Further run details are still being documented. Proofs will be published on GitHub.
Model and setup labels describe how each result was obtained; they are not a controlled comparison of approaches.
What do the human contribution classes mean?
One-prompt
The model worked independently after the initial prompt, with only continuation requests.
Five-prompts
A collaborator supplied substantive questions, suggestions, or requests.
Hard
The model mainly followed the researcher’s direction.
N/A
Not classified, or this classification does not apply.
These labels describe guidance during the investigation, including partial results and proposals awaiting review. They do not describe the effort spent checking a proof.
What do the verification statuses mean?
Editor-accepted
Accepted by the Kourovka editors as reported by the project team, or listed as a solution in their official repository. This is distinct from journal refereeing.
Team-checked
Verified by the project at the stated scope. This status does not imply editorial acceptance, historical originality, or formal verification.
Partial progress
Verified results for specified cases or a qualified formulation. The full problem remains unresolved; the scope is stated in the record.
Pending review
A proposed solution is awaiting checks by the project team.
Solved elsewhere
An existing published or editor-recognized answer identified by the project. Authorship is credited in each summary; screening an existing result is not a new project solution.
Preprint
A paper posted to arXiv by its authors and not yet refereed. This status is separate from the verification register for problem records.
03 / TEAM
Our team
We bring together experience in mathematics, AI, and mathematical education.
Affiliations
Prof. Elena Bunina
Bar-Ilan University
Prof. Alexei Miasnikov
Stevens Institute of Technology
Prof. Efim Zelmanov
Fields Medal, 1994
Southern University of Science and Technology, China University of California, San Diego, USA (emeritus)
Prof. Evgeny Khukhro
Co-editor of the Kourovka Notebook
School of Engineering and Physical Sciences, University of Lincoln, UK (emeritus)
Vlad Stepanov
AI in Math & Math Education Lab, Stevens Institute of Technology
Stan Fedotov, PhD
Nebius Academy
Maria Matveeva
AI in Math & Math Education Lab, Stevens Institute of Technology
Kyrylo Muliarchyk, PhD
Bar-Ilan University
Students
About ten master’s and PhD students from Bar-Ilan University, MIT, and Stevens Institute of Technology take part in the research.
We invite researchers, teachers, and students to discuss and try new ways of doing mathematics and teaching it.
We are interested in approaches to research and education that develop all five roles, take the real difficulties of this change seriously, and preserve space for deep, independent thought.
