{
  "abstract": "We prove that if H and K are subgroups of a free group F of arbitrary rank and $H'=K'\\ne 1$, then $H=K$. As a consequence, if R is a noncyclic subgroup of a free group F and $R'$ is normal in F, then R is normal in F. This gives a positive answer to a problem of Shpilrain, originally posed as Problem 11.124 in the Kourovka Notebook and later appearing as Problem F15 in Kapovich-Myasnikov-Shpilrain.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06755",
  "author_total": 1,
  "authors": [
    {
      "name": "Artur Tursunbaev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a28gdb1da1qp2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06755",
  "fulltext_url": null,
  "id": "a28gdb1da1qp2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06755",
  "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",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7212025124",
  "pdf_url": "https://arxiv.org/pdf/2609.06755",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a28gdb1da1qp2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.06755",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06755"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Subgroups of free groups with the same derived subgroup",
  "title_text": "Subgroups of free groups with the same derived subgroup",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/a28gdb1da1qp2",
  "venue": "arXiv",
  "versions": []
}