{
  "abstract": "Let $R$ be a nonzero commutative local ring with $2,3\\in R^\\times$, and let $ρ\\in\\Aut(R)$ satisfy $ρ^3=\\id$. We classify abstract automorphisms of the elementary and full triality groups in the adjoint and simply connected forms of type $D_4$. Every automorphism is uniquely determined by an adjoint group element and a ring automorphism preserving the trace $T=\\id+ρ+ρ^2$; conversely, every such pair defines an automorphism. A square-zero example shows that trace-preserving coefficient automorphisms need not normalize $\\langleρ\\rangle$ when $ρ$ is nontrivial on $R$ but trivial on its residue field. We also prove a relative rigidity theorem for homomorphisms from elementary triality groups over arbitrary coefficient rings to local targets with $2$ and $3$ invertible, assuming that an adjoint conjugation makes the parameter-one root images standard modulo the maximal ideal. Conversely, every unital trace-preserving ring homomorphism into such a local coefficient ring induces the corresponding homomorphism of elementary groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21662",
  "author_total": 3,
  "authors": [
    {
      "name": "Elena Bunina"
    },
    {
      "name": "Avi Berezovskiy"
    },
    {
      "name": "Deep Makadiya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6rv4khaph34xr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21662",
  "fulltext_url": null,
  "id": "6rv4khaph34xr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21662",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7222480955",
  "pdf_url": "https://arxiv.org/pdf/2610.21662",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6rv4khaph34xr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21662",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21662"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Automorphisms of triality groups over local rings",
  "title_text": "Automorphisms of triality groups over local rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/6rv4khaph34xr",
  "venue": "arXiv",
  "versions": []
}