A classification of skew morphisms of dihedral groups

Abstract

Abstract A skew morphism of a finite group 𝐴 is a permutation 𝜑 of 𝐴 fixing the identity element and for which there is an integer-valued function 𝜋 on 𝐴 such that φ ⁢ ( x ⁢ y ) = φ ⁢ ( x ) ⁢ φ π ⁢ ( x ) ⁢ ( y ) \varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y) for all x , y ∈ A x,y\in A . In this paper, we restrict ourselves to the case when A = D n A=D_{n} , the dihedral group of order 2 ⁢ n 2n . Wang et al. [Smooth skew morphisms of dihedral groups, Ars Math. Contemp. 16 (2019), 2, 527–547] determined all 𝜑 under the condition that π ( φ ( x ) ) ≡ π ( x ) ( mod | φ | ) ) \pi(\varphi(x))\equiv\pi(x)\pmod{\lvert\varphi\rvert}) holds for every x ∈ D n x\in D_{n} , and later Kovács and Kwon [Regular Cayley maps for dihedral groups, J. Combin. Theory Ser. B 148 (2021), 84–124] characterised those 𝜑 such that there exists an inverse-closed ⟨ φ ⟩ \langle\varphi\rangle -orbit, which generates D n D_{n} . We show that these two types of skew morphisms comprise all skew morphisms of D n D_{n} . The result is used to classify the finite groups with a complementary factorisation into a dihedral and a core-free cyclic subgroup. As another application, a formula for the total number of skew morphisms of D p t D_{p^{t}} is also derived for any prime 𝑝.