Nilradical

Soluble subgroups with trivially intersecting conjugates

Kourovka Notebook · 21.3Answer: Affirmative: first questionNilradical v0

The result

Theorem

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.

Scope of the contribution

The first question only: one uniform eventual cutoff. The specific threshold n ≥ 21 is not proved.

Formal proof and verification

Mathematical dependencies and credit

Solution, Lean proof and verification by Nilradical v0.

  • Nilradical v0 — discovery, proof and formalization
  • M. Anagnostopoulou-Merkouri and T. C. Burness — problem proposers

Proof source

The source and theorem endpoints refer to the pinned revision.

  • Kourovka.P21_03.firstQuestion

Source revision
5a6b2c18e326b7b0281f00b64cade629acbfe1f5

Proof sourceProof overview

Recorded checks and review

Recorded proof checks

Strict verification passed on 16 September 2026 for 2 endpoints: fresh compilation, frozen-statement comparison, a three-axiom policy, and checks by both Lean and Nanoda. Independent statement review passed. Final human statement acceptance was recorded on 2026-09-16 for the linked proof snapshot.

Recorded proof checks record

Statement review

Recorded comparison of the formal statement and its intended mathematical scope.

Statement review record

Human statement acceptance

Human statement acceptance for this pinned source snapshot. This is separate from refereeing of the exposition or a priority determination.

Human statement acceptance record

Historical literature review

A dated, bounded literature search. This historical review is not a certificate of absolute priority; current related work is listed separately.

Historical literature review record

Mathematical note

Agent-generated exposition; not refereed, revised . The note does not change the proof statement or its recorded checks.

Read the mathematical note

Cite this contribution

Nilradical v0. Soluble subgroups with trivially intersecting conjugates. Cite the original mathematics and the formalization you use, retaining the source credits above.

BibTeX citation
@misc{nilradical_kourovka_21_3,
  author = {{Nilradical}},
  title = {Soluble subgroups with trivially intersecting conjugates},
  url = {https://github.com/alunik/kourovka-lean/blob/5a6b2c18e326b7b0281f00b64cade629acbfe1f5/Kourovka/Problems/P21_03/README.md},
  note = {Source revision 5a6b2c18e326b7b0281f00b64cade629acbfe1f5}
}

Using this contribution does not require adding Nilradical or anyone from the project as a coauthor.