A classification of skew morphisms of dihedral groups

A skew morphism of a finite group A is a permutation phi of A fixing the identity element and for which there is an integer-valued function pi on A such that phi(xy) = phi(x)phi(pi(x))(y) for all x, y is an element of A. In this paper, we restrict ourselves to the case when A = D-n, the dihedral group of order 2n. Wang et al. [Smooth skew morphisms of dihedral groups, Ars Math. Contemp. 16 (2019), no. 2, 527-547] determined all phi under the condition that pi(phi(x)) equivalent to pi(x) (mod vertical bar phi vertical bar)) holds for every x is an element of D-n, and later Kovacs and Kwon [Regular Cayley maps for dihedral groups, J. Combin. Theory Ser. B 148 (2021), 84-124] characterised those phi such that there exists an inverse-closed <phi >-orbit, which generates D-n. We show that these two types of skew morphisms comprise all skew morphisms of 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-pt is also derived for any prime p.