{
  "abstract": "Let $X$ be a finite skew brace, and let $L_\\infty(X)$ be the final term of its left series. We introduce the left-nilpotent residual $\\RL(X)$ and prove that $ \\RL(X)=\\Id_X\\bigl(L_\\infty(X)\\bigr). $ Thus $X/\\RL(X)$ is the largest left-nilpotent quotient of $X$. For skew braces of nilpotent type, we relate $\\RL(X)$ to $γ_\\infty(X,\\cdot)$ and obtain conditions ensuring that $L_\\infty(X)$ is an ideal, including the case where $|X|$ is cube-free. Our main results concern supplements to $\\RL(X)$. We prove existence results using coprime action and Sylow and Hall theory, and give an example of order $18$ whose residual has no proper supplement. Finally, iterating $\\RL$ yields the largest perfect subskew brace of $X$, and $X$ is KSV-solvable if and only if this subskew brace is zero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06994",
  "author_total": 2,
  "authors": [
    {
      "name": "Gülin Ercan"
    },
    {
      "name": "İsmail Ş. Güloğlu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9k1b396wd4xc6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06994",
  "fulltext_url": null,
  "id": "9k1b396wd4xc6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06994",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7211987913",
  "pdf_url": "https://arxiv.org/pdf/2609.06994",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9k1b396wd4xc6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06994",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06994"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Left-Nilpotent Residual and Its Supplements in Finite Skew Braces",
  "title_text": "The Left-Nilpotent Residual and Its Supplements in Finite Skew Braces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/9k1b396wd4xc6",
  "venue": "arXiv",
  "versions": []
}