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Soluble subgroup pairs in symmetric and alternating groups

A mix of human-written and AI-generated textHuman understanding: some partsmath.GR — Group Theory

Contributed by Qiyue Tang ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

We prove that every pair of soluble subgroups of the symmetric group SnS_n or the alternating group AnA_n has trivially intersecting conjugates whenever n≥21n\ge21. This answers the second question of Anagnostopoulou-Merkouri and Burness in Problem 21.3 of the Kourovka Notebook.

Provenance statement

The proof was developed using ChatGPT-6-Pro during a research conversation lasting approximately 157 minutes, as recorded in the research package. AI contributed to the mathematical arguments, computational methods, and internal review. Codex was subsequently used to audit and revise the manuscript, clarify definitions and proof details, and reproduce the computational checks. The work also underwent external AI-assisted review using Rethlas-OSV; the certificate-decoding documentation issue identified in the supplied report was corrected. The computations use exact integer and rational arithmetic in Python, permutation-group calculations in GAP, and graph-automorphism calculations with nauty/Traces. The certificates, source code, reproduction instructions, and verification records are publicly available at https://github.com/tqyqiqi1509/kourovka-21-3b-certificates.

References

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  1. V. I. Zenkov, On intersections of nilpotent subgroups in finite symmetric and alternating groups, Proc. Steklov Inst. Math. 285 (2014), suppl. 1, S203--S208, doi:10.1134/S0081543814050228 (https://doi.org/10.1134/S0081543814050228).DOI
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Version history

  1. v1Submitted by Qiyue TangInitial depositCurrentNov 03, 2026