Reciprocal skew morphisms of cyclic groups

Abstract

A skew morphism of the cyclic additive group Z_n is a bijection \phi on Z_n forwhich there exists an integer function \pi : Z_n \to Z such that \phi(0) = 0 and \phi(x +y) = \phi(x) + \phi^{\pi(x)}(y) for all x, y \in Z_n. A pair of skew morphisms \phi: Z_n \to Z_nand \phi^{\star}: Z_m \to Z_m are reciprocal if (a) the orders of \phi and \phi^{\star} divide m and n,respectively, and (b) the associated power functions \pi and \pi^{\star} are determined by\pi (x) = \phi^{\star x}(1) and \pi^{\star}(y) = \phi^y(1). Pairs of reciprocal skew morphisms of cyclicgroups are in one-to-one correspondence with isomorphism classes of regular dessinswith complete bipartite underlying graphs. In this paper we determine all pairs ofreciprocal skew morphisms of the cyclic groups provided that one of them is a groupautomorphism.