On the skew-morphisms of dihedral groups

Abstract

Abstract A skew-morphism φ of a finite group G is a permutation on G such that φ ⁢ ( 1 ) = 1 ${\varphi(1)\hskip-0.853583pt=\hskip-0.853583pt1}$ and φ ⁢ ( g ⁢ h ) = φ ⁢ ( g ) ⁢ φ π ⁢ ( g ) ⁢ ( h ) ${\varphi(gh)=\varphi(g)\varphi^{\pi(g)}(h)}$ for all g , h ∈ G ${g,h\in G}$ , where π is a function from G to 𝐙 | φ | ${\mathbf{Z}_{|\varphi|}}$ , called the power function of φ. Furthermore, we say φ is of skew-type k, provided π takes on exactly k values in 𝐙 | φ | ${\mathbf{Z}_{|\varphi|}}$ , and call the set Ker ⁡ φ = { g ∈ G ∣ π ⁢ ( g ) = 1 } ${\operatorname{Ker}\varphi=\{g\in G\mid\pi(g)=1\}}$ the kernel of φ, which is actually a subgroup of G (see [6]). Though skew-morphism is a pure group theoretical concept, it is closely related to regular Cayley maps. We have a long term goal to classify all regular Cayley maps for dihedral groups. In this paper, we mainly work on the skew-morphisms of dihedral groups. Let φ be a skew-morphism of the dihedral groups D 2 ⁢ n = 〈 a , b ∣ a n = b 2 = ( a ⁢ b ) 2 = 1 〉 ${D_{2n}=\langle a,b\mid a^{n}=b^{2}=(ab)^{2}=1\rangle}$ . We prove that the kernel of φ is contained in the cyclic subgroup 〈 a 〉 ${\langle a\rangle}$ if and only if φ is of skew-type 4 and preserves 〈 a 〉 ${\langle a\rangle}$ . In the case of φ preserving 〈 a 〉 ${\langle a\rangle}$ , we give some characterizations of φ, in particular, we prove that φ is of skew-type 1, 2 or 4. Working from these characterizations, an infinite family of skew-morphisms of D 2 ⁢ n ${D_{2n}}$ are constructed, all of which are of skew-type 4 and have kernel 〈 a 2 〉 ${\langle a^{2}\rangle}$ , which gives a positive answer to an open problem raised recently by M. Conder [1].