{
  "abstract": "Throughout this paper, all groups are finite. Let $\\mathrm{form}(G)$ denote the formation generated by a group $G$. A formation $\\mathfrak{F}$ is said to be $S_n$-closed if every subnormal subgroup of every group in $\\mathfrak{F}$ belongs to $\\mathfrak{F}$. A. N. Skiba and L. A. Shemetkov posed the problem of proving that $\\mathrm{form}(G)$ has only finitely many $S_n$-closed subformations for every group $G$ (Problem 14.81 of the Kourovka Notebook). In this paper we propose a negative answer to this problem. We prove that for the group $G_0=(\\mathrm{SL}_2(5)\\rtimes C_2)\\times\\mathrm{SL}_2(5)$ of order 28800, where $C_2$ acts on $\\mathrm{SL}_2(5)$ by an outer automorphism, the formation $\\mathrm{form}(G_0)$ has an infinite strictly descending chain of $S_n$-closed subformations. The main tool is a family of central-valued multilinear maps, defined for every group in $\\mathrm{form}(G_0)$ by its outer actions on its components; their vanishing is preserved under taking epimorphic images, subdirect products and subnormal subgroups. The members of the chain are separated by central quotients of the groups $\\mathrm{SL}_2(5)^{2^d-1}\\rtimes C_2^d$, $d\\geq 2$. Using Burichenko's description of the lattice of subformations of $\\mathrm{form}(\\mathrm{SL}_2(5)\\rtimes C_2)$, we also prove that every non-empty $S_n$-closed subformation of $\\mathrm{form}(G_0)$ is generated, as an ordinary formation, by a single group, and that the set of such subformations is countably infinite. Finally, we determine the intersection of the chain and give an explicit generator of it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37462",
  "author_total": 1,
  "authors": [
    {
      "name": "Chenxiao Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9916tnpps3qep.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37462",
  "fulltext_url": null,
  "id": "9916tnpps3qep",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37462",
  "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"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7224082903",
  "pdf_url": "https://arxiv.org/pdf/2610.37462",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9916tnpps3qep.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37462",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37462"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a problem of A. N. Skiba and L. A. Shemetkov concerning $S_n$-closed subformations of a one-generated formation",
  "title_text": "On a problem of A. N. Skiba and L. A. Shemetkov concerning S_n-closed subformations of a one-generated formation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/9916tnpps3qep",
  "venue": "arXiv",
  "versions": []
}