Smooth skew morphisms on semi-dihedral groups (Q6639851)

A \emph{skew morphism} of a finite group \(G\) is a bijection \(\varphi : G \to G\) such that \(\varphi(1) = 1\), and there exists a function \(\pi: G \to \mathbb{Z}_{\mathrm{ord}(\varphi)}^\times\) such that,\N\[\N\text{for every } g, h \in G , \quad \varphi(gh) = \varphi(g) \varphi^{\pi(g)}(h) \,.\N\]\NThis concept has proven surprisingly useful in addressing various combinatorial questions. Indeed, it was introduced in [\textit{R. Jajcay} and \textit{J. Širáň}, Discrete Math. 244, No. 1--3, 167--179 (2002; Zbl 0988.05047)], where it was observed that, for a \emph{regular Cayley map} \(\mathrm{Cay}(G, S, \rho)\), the local ordering \(\rho\) is the restriction of a skew morphism \(\varphi\) to \(S\), thereby linking the existence of such a graph embedding to the existence of \(\varphi\). Subsequently, this concept formed the basis for the classification of Cayley maps with \emph{two orbits on darts} [\textit{R. Jajcay} and \textit{R. Nedela}, Graphs Comb. 31, No. 4, 1003--1018 (2015; Zbl 1317.05036)] and for the study of \emph{factorizations of finite groups} as products of a monolithic and a cyclic subgroup [\textit{M. Bachratý} et al., Algebr. Comb. 5, No. 5, 785--802 (2022; Zbl 1517.20029)].\N\NFinding skew morphisms is a significant challenge, as they carry substantial algebraic and combinatorial information about the underlying group. As an emblematic example, even the skew morphisms of cyclic groups have not yet been classified [\textit{M. Bachratý} and \textit{M. Hagara}, J. Algebr. Comb. 61, No. 1, Paper No. 5, 23 p. (2025; Zbl 07960247)]. To address this, it was proposed to focus on more manageable classes of skew morphisms. For instance, in [\textit{M. Bachratý} and \textit{R. Jajcay}, Australas. J. Comb. 67, Part 2, 259--280 (2017; Zbl 1375.05126)], the study of smooth skew automorphisms was initiated: a skew automorphism \(\varphi\) is \emph{smooth} if \(\pi^{-1}(1)\) is \(\varphi\)-invariant. The smoothness hypothesis simplifies the structure of skew morphisms considerably: \(\pi: G \to \mathbb{Z}_{\mathrm{ord}(\varphi)}^\times\) becomes a group homomorphism, and \(\varphi\) induces the identity map on the factor group \(G / \pi^{-1}(1)\).\N\NThis paper under review provides a complete classification of smooth skew morphisms on semidihedral groups, defined by the presentations\N\[\NSD_{4n} = \langle a, b \mid a^{4n} = b^2 = a^b a^{1-2n} = 1 \rangle,\quad\text{for } n \geq 3 \,.\N\]\NSection~2 presents the updated toolbox for the study of smooth skew morphisms, largely based on [\textit{N.-E. Wang} et al., Ars Math. Contemp. 16, No. 2, 527--547 (2019; Zbl 1416.05302)]. A novel observation is that \([G, G]\) is a subgroup of \(\pi^{-1}(1)\), which enables the authors to enumerate all possible subgroups of \(SD_{4n}\) that could serve as candidates for \(\pi^{-1}(1)\). In Sections~3 and~4, the classification is completed through a detailed and technical analysis of all skew morphisms arising from each case in the list of possible \(\pi^{-1}(1)\).