Abstract:Let $n, m \ge 4$. We classify the Zappa--Szép products $G = HK$ with $H = \langle x\rangle \rtimes \langle y\rangle \cong \mathrm{SD}_{2^n}$ and $K = \langle z\rangle \rtimes \langle w\rangle \cong \mathrm{SD}_{2^m}$, 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 obtain a complete classification of such exact products by 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 $[x, z] = 1$ 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. In contrast with the dihedral case, we further construct an explicit exact product with both cores non-trivial and $[x, z] \ne 1$, showing that the parameter space in the semidihedral setting is strictly richer than its dihedral analogue.
| Subjects: | Group Theory (math.GR) |
| MSC classes: | 20D40, 20D60, 20F05, 20D15, 20E22 |
| Cite as: | arXiv:2605.24480 [math.GR] |
| (or arXiv:2605.24480v1 [math.GR] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24480 arXiv-issued DOI via DataCite (pending registration) |
From: Riccardo Aragona [view email]
[v1]
Sat, 23 May 2026 09:06:43 UTC (19 KB)
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