{
  "abstract": "A group (G,*) is natural if its group operation is uniquely determined by a metric structure (G,d) on G in the following sense: every group structure (G,.) for which all right translations are isometries of (G,d) must be isomorphic to (G,*). Examples included all connected Lie groups or all groups generated by involutions (Sutherland). The orientation rigidity theorem of Leemann and de la Salle allows to upgrade the non-abelian structure theorem: every non-abelian group of cardinality not larger than the continuum that is not generalized dicyclic is natural. A consequence is that all non-metabelian Lie groups, all simple group of cardinality not larger then the continuum, all homeomorphism-, diffeomorphism -or symplectomorphism groups of manifolds, or automorphism groups probability spaces or non-abelian crystallographic groups are natural. Also the abelian structure theorem is extended: affine rigidity of Jarosz, together with Mazur-Ulam's theorem implies that the additive group of every real or complex vector spaces is natural and that all connected abelian Banach Lie groups are natural. A theorem of Babai implies that every Boolean group is natural for every cardinality. Undecided is whether C_p^k is natural for cardinals k>c and odd prime p, and whether the additive group of any field F is natural, or whether the cardinality assumption in the non-abelian structure theorem is needed. A major question is whether G^2=G implies that G is natural. This is already open for abelian groups: does 2G=G imply that G is natural?",
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  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07677",
  "author_total": 1,
  "authors": [
    {
      "name": "Oliver Knill"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtg0b38gh5exe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07677",
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  "landing_url": "https://arxiv.org/abs/2609.07677",
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  "openalex": "W7212023867",
  "pdf_url": "https://arxiv.org/pdf/2609.07677",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtg0b38gh5exe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07677",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07677"
    }
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Natural Groups",
  "title_text": "On Natural Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/dtg0b38gh5exe",
  "venue": "arXiv",
  "versions": []
}