What should mathematics become with AI?

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.

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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.

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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.

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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: Team-checked. Model and setup attribution supplied by the project team.

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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.

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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: Team-checked. Model and setup attribution supplied by the project team.

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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.

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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: Team-checked. Model and setup attribution supplied by the project team.

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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.

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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: 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.

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A finite group G = ⟨A, B⟩ with d(A) = d(B) = 2, coprime orders |A| and |B|, and d(G) = 4 gives a negative answer already for d = 2. The subgroups live in a direct power of PSL(2, 11); the proof combines exact finite counts in this group of order 660 with Lagrange’s theorem and the fact that a homomorphism is determined by its values on generators. Two archived checkers agree, but an independent GAP reproduction is still pending.

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.

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Over an explicit finitely presented noncommutative algebra R of characteristic two, a matrix σ in GL₃(R) has [[σ, T₁], T₂] of infinite order for every pair of nonidentity matrices with (Tᵢ − I)² = 0, so no such double commutator is unipotent. This refutes the statement for noncommutative R; it makes no claim about commutative rings or rings in which 2 is invertible. The proof rests on Gerasimov’s theorem on units of ring coproducts and a free-product tree argument, without finite enumeration.

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.

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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.

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The proposed analogue fails for every rank n ≥ 2: explicit examples have n + 1 successive orbit elements generating the free group, while the first n generate a subgroup of infinite index. The write-up also gives counterexamples to a normal-closure variant at rank 2 and a subset variant at rank 3.

Review status: Team-checked. Model and setup attribution supplied by the project team.

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For the explicit integer N = 2¹²⁴²¹⁰³⁵⁰, a finite 2-group of order 2¹¹⁰ᴺ⁺¹ has every abelian normal subgroup generated by at most 32N + 1 elements, while it contains an elementary abelian subgroup requiring 65N generators. This gives a negative answer to Pyber’s question. The group multiplication is given by finite deterministic formulas, and the proof is self-contained without enumeration of the group or its subgroups.

Review status: Team-checked. Model and setup attribution supplied by the project team.

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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: Team-checked. Model and setup attribution supplied by the project team.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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: 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.

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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.

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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.

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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: 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.

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An explicit finite 2-group has all abelian normal subgroups of order at most 2⁴²ᵉ, but an elementary abelian subgroup of order 2⁸⁵ᵉ. This refutes the proposed bound. The construction is specified mathematically without enumerating the enormous group.

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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