Finitely presented infinite $p$-groups for every prime

As the paper states it, in briefFor every prime $p$, an infinite, residually finite group with an ordinary finite presentation in which every element has finite $p$-power order. One presentation, written over the integers, serves every prime.

• Extended Can an infinite periodic group have an ordinary finite presentation? posed by others
• Answered Is there a finitely presented infinite $p$-group for every prime?

Blind checks

A blind check by a model from another developer

Audits

A line-by-line review.

Exact certificates

Exact mod-$p$ checks in the repository

Lean

None.

Peer review

Not yet.

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Label checked 30 October 2026.

Novelty

Concurrent with named independent work: Kenta Kitamura proved the every-prime result independently (manuscript of 13 October 2026, Lean formalization of 16 October 2026); in the paper’s words, “the extension to every prime listed above is shared with it.” It extends OpenAI’s construction for $p=2$, whose conceptual credit is theirs; a lemma is credited to Zalesskii (1985).

Correction

Version 1.0.1 adds a note on Kenta Kitamura’s independent, concurrent work, which version 1.0.0 did not cite. Corrections

PDF

papers/p-groups-v1.0.1.pdf, version 1.0.1, 715,241 bytes, 38 pages.

SHA-256 23b1d37fb195261a3394bd8899c8e50811b690588c2f3c47cf295290414b644a

DOI

10.5281/zenodo.23791059; all versions 10.5281/zenodo.23401290

Code

github.com/Apsiape/finitely-presented-p-groups

1. 30 Oct 2026Version 1.0.1. Credit: a “Note added” in the introduction cites Kenta Kitamura’s independent, concurrent work (manuscript 13 October 2026, Lean 16 October 2026), which also proves the every-prime result; in the paper’s words, “the extension to every prime listed above is shared with it.”doi:10.5281/zenodo.23791059

All corrections, and what counts as one.

@misc{douglas2026pgroups,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {Finitely presented infinite p-groups for every prime},
  year    = {2026},
  version = {1.0.1},
  doi     = {10.5281/zenodo.23791059},
  url     = {https://doi.org/10.5281/zenodo.23791059}
}

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