On Zappa-Szép products of two semidihedral groups

Abstract

Let n, m ≥ 4. We study the Zappa--Szép products G = HK with H = ⟨ x⟩ rtimes ⟨ y⟩ ≅ SD_2ⁿ and K = ⟨ z⟩ rtimes ⟨ w⟩ ≅ 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 ⟨ x⟩ and ⟨ z⟩ in~G. First, when both ⟨ x⟩ and ⟨ z⟩ 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 ⟨ x⟩_G and ⟨ z⟩_G are arbitrary subgroups of ⟨ x⟩ and ⟨ z⟩, under the simplifying assumption [x, z] = 1, and for generating systems for which all mixed commutators lie in ⟨ x⟩⟨ z⟩, 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] ≠ 1, so that the semidihedral setting contains additional families of exact products.

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BibTeX RIS

BibTeX

@misc{aragona2026zappaszép,
  title = {{On Zappa-Szép products of two semidihedral groups}},
  author = {Riccardo Aragona},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2605.24480},
  eprint = {2605.24480},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.24480},
}

RIS

TY  - UNPB
TI  - On Zappa-Szép products of two semidihedral groups
AU  - Riccardo Aragona
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2605.24480
UR  - https://arxiv.org/abs/2605.24480
AB  - Let n, m ≥ 4. We study the Zappa--Szép products G = HK with H = ⟨ x⟩ rtimes ⟨ y⟩ ≅ SD_2^n and K = ⟨ z⟩ rtimes ⟨ w⟩ ≅ 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 ⟨ x⟩ and ⟨ z⟩ in~G. First, when both ⟨ x⟩ and ⟨ z⟩ 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 ⟨ x⟩_G and ⟨ z⟩_G are arbitrary subgroups of ⟨ x⟩ and ⟨ z⟩, under the simplifying assumption [x, z] = 1, and for generating systems for which all mixed commutators lie in ⟨ x⟩⟨ z⟩, 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] ≠ 1, so that the semidihedral setting contains additional families of exact products.
ER  -

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