Decomposition of skew-morphisms of cyclic groups

Abstract

A skew-morphism of a group H is a permutation σ of its elements fixing the identity such that for every x, y ∈ H there exists an integer k such that σ(xy)=σ(x)σ^k(y). It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups Z_n: if n = n₁n₂ such that (n₁, n₂) = 1, and (n₁, φ(n₂)) = (φ(n₁), n₂) = 1 (φ denotes Euler's function) then all skew-morphisms σ of Z_n are obtained as σ = σ₁ × σ₂, where σ_i are skew-morphisms of Z_{n_i}, i = 1, 2. As a consequence we obtain the following result: All skew-morphisms of Z_n are automorphisms of Z_n if and only if n = 4 or (n, φ(n)) = 1.