{
  "abstract": "Let $r>1$ with $\\gcd(r,30)=1$. We construct a finite group $G_r=(C_5\\times C_3\\times C_r)\\rtimes W$, where $|W|=32$ and $G_r'\\cong C_{60r}$, together with an augmentation-preserving automorphism $α_r\\in\\operatorname{Aut}(\\mathbb{Z}G_r)$ having no Zassenhaus factorization. The image $Y_r=α_r(G_r)$ is a normalized group basis which is not rationally conjugate to $G_r$, although every element of $Y_r$ is individually rationally conjugate to an element of $G_r$. Consequently, (ZC2) and (ZC3) fail for finite cyclic-by-abelian groups, resolving a problem of Margolis and del Río. The construction extends Hertweck's example uniformly: both the class-preserving obstruction and the integral gluing are independent of the order of the auxiliary factor. The smallest admissible member of this family has order $3360$ and derived subgroup $C_{420}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.03254",
  "author_total": 1,
  "authors": [
    {
      "name": "Brecht F. Verbeken"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99f4nbkach1te.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.03254",
  "fulltext_url": null,
  "id": "99f4nbkach1te",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.03254",
  "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",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7172523604",
  "pdf_url": "https://arxiv.org/pdf/2608.03254",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99f4nbkach1te.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2608.03254",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.03254"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Cyclic-by-abelian counterexamples to the second and third Zassenhaus conjectures",
  "title_text": "Cyclic-by-abelian counterexamples to the second and third Zassenhaus conjectures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/99f4nbkach1te",
  "venue": "arXiv",
  "versions": []
}