Nilradical
A mathematical research agent with answers to eight Kourovka Notebook problems in group theory. Each has a complete Lean 4 proof replayed by an independent checker, and a statement accepted by a human reviewer. The results are for mathematicians to check, develop and publish as their own work. The agent asks for a citation, not coauthorship.
Results
Affirmative and negative answer the question as posed.
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16.68
Surjectivity of word maps on PSL(2,C) and PSL(2,R)
Every nonidentity word in two variables induces a surjective map on PSL(2,C); an explicit word of length 44 induces a map on PSL(2,R) that misses every nonidentity involution and so is not surjective.
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21.3
Soluble subgroups with trivially intersecting conjugates
For every sufficiently large degree, any two soluble subgroups H and K of the symmetric or alternating group have a conjugate of K that meets H trivially.
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21.35
A p-nilpotency criterion for multilinear commutator words
For a finite group G, a prime p and a multilinear commutator word w, the verbal subgroup w(G) has a normal p-complement if and only if p divides |xy| whenever x and y are w-values with |x| prime to p and p dividing |y|.
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21.38
A subgroup of Thompson's group F with spread exactly one
The subgroup of Thompson's group F consisting of maps with equal slopes at the two endpoints is infinite and has spread exactly one: every nonidentity element has a generating companion, but some two elements share none.
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21.44
A dense two-generator subgroup of subexponential growth
Two explicit automorphisms of the rooted five-ary tree generate a dense subgroup of subexponential growth in the inverse limit of the iterated wreath products of the alternating group of degree five.
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21.53
Product orders 2 and 3 of involutions do not determine the rest
In a finite nonabelian simple matrix group over GF(8), swapping two involutions of a conjugacy class and fixing the others preserves every product of order 2 or 3 but sends a product of order 5 to one of order 7.
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21.68
A semiabelian group with a nonmonomial irreducible character
A semiabelian group of order 2592 has an irreducible complex character of degree 8 that is not induced from a linear character of any subgroup.
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21.106
A first-order formula that is not concise in residually finite groups
A positive, parameter-free, one-variable first-order formula takes exactly two values in the integral Heisenberg group, and those two values generate an infinite cyclic subgroup.
How a result is produced and checked
- Novelty. A dated literature search confirms the question is open; it is refreshed before release.
- Exploration and falsification. Candidate ideas are developed, then attacked with exact computation in search of counterexamples.
- Written proof and review. Independent reviewers check a complete written argument, its hypotheses and its use of earlier work.
- Formalization and audit. The claim is formalized in Lean 4; statement and axioms are audited, and the proof is replayed by an independent checker.
- Human acceptance. A person checks the final Lean statement against the original question and accepts the result.
- Note. The agent then writes a short mathematical note on the result and its proof, informal and not refereed.
About the agent
Nilradical v1.0.0 uses Codex and GPT Pro, GAP and Magma, Lean 4 and mathlib, and the independent checker Nanoda. github.com/alunik/nilradical.
Using and citing results
You do not need permission, and you do not need to add Nilradical, its developer or project team as coauthors. A shorter or clearer proof is a valuable contribution.
Cite the result and the earlier work you use. Records give BibTeX for exact source revisions; the citation file (CFF) covers the agent. No DOIs have been minted.