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A universal group with thirteen relators

Primarily AI-generated textHuman understanding: some partsmath.GR — Group Theory

Contributed by Qiuyu Ren ↗

Version 1 / Oct 16, 2026 / CC BY 4.0

Abstract

We construct a finitely presented group with 13 relators that contains an isomorphic copy of every finitely presented group. This improves Valiev's bound of 21 relators for universal groups. The proof combines Valiev's construction, subgroup calculations of Kegel--Li--Ren, and small-cancellation theory.

Provenance statement

GPT-6 Astra was used to generate the proof, the writeup, and a formalization of the proof in Lean 4.

Tools used

OpenAI
ChatGPTVersion GPT-6 Astra

References

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Show all 12 referencesShow fewer references
  1. Q. Ren, Universal-Group , Lean 4 formalization, GitHub repository (https://github.com/32805433/Universal-Group), 2026, commit a691301 (https://github.com/32805433/Universal-Group/tree/a6913011408330cc2039d3a9b95daf14f676f405).link
  2. M. K. Valiev, Universal group with twenty-one defining relations , Discrete Math. 17 (1977), 207--213. doi:10.1016/0012-365X(77)90153-4 (https://doi.org/10.1016/0012-365X(77)90153-4).DOI

Version history

  1. v1Submitted by Qiuyu RenInitial depositCurrentOct 16, 2026