{
  "abstract": "Let $n, m \\ge 4$. We study the Zappa--Szép products $G = HK$ with $H = \\langle x\\rangle \\rtimes \\langle y\\rangle \\cong SD_{2^n}$ and $K = \\langle z\\rangle \\rtimes \\langle w\\rangle \\cong SD_{2^m}$, together with the fixed generators $x, y, z, w$, and we describe the parameter tuples which encode the mixed commutators of these generators, according to the cores of $\\langle x\\rangle$ and $\\langle z\\rangle$ in~$G$. First, when both $\\langle x\\rangle$ and $\\langle z\\rangle$ are normal in~$G$, we show that the admissible parameter tuples are exactly the solutions of an explicit system of six polynomial congruences. Second, when the cores $\\langle x\\rangle_G$ and $\\langle z\\rangle_G$ are arbitrary subgroups of $\\langle x\\rangle$ and $\\langle z\\rangle$, under the simplifying assumption $[x, z] = 1$, and for generating systems for which all mixed commutators lie in $\\langle x\\rangle\\langle z\\rangle$, we obtain an analogous classification by twelve congruences together with two order conditions; this is the semidihedral counterpart of the Hu--Yu classification~\\cite{HuYu2025} for dihedral groups $D_{2n}$ and $D_{2m}$ with $n, m$ odd. In contrast with that case, we further construct an explicit exact product with both cores non-trivial and $[x, z] \\ne 1$, so that the semidihedral setting contains additional families of exact products.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.24480",
  "author_total": 1,
  "authors": [
    {
      "name": "Riccardo Aragona"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9zwhhs7sbz549.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.24480",
  "fulltext_url": null,
  "id": "9zwhhs7sbz549",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.24480",
  "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": "W7162421916",
  "pdf_url": "https://arxiv.org/pdf/2605.24480",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9zwhhs7sbz549.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:28Z",
      "native_id": "2605.24480",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.24480"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Zappa-Szép products of two semidihedral groups",
  "title_text": "On Zappa-Szép products of two semidihedral groups",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/9zwhhs7sbz549",
  "venue": "arXiv",
  "versions": []
}