### 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*=*n*_{1}*n*_{2}such that (*n*_{1},*n*_{2}) = 1, and (*n*_{1},*φ*(*n*_{2})) = (*φ*(*n*_{1}),*n*_{2}) = 1 (*φ*denotes Euler's function) then all skew-morphisms*σ*of Z_{n}are obtained as*σ*=*σ*_{1}×*σ*_{2}, 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.#### Keywords

Cyclic group, permutation group, skew-morphism, Schur ring

DOI: https://doi.org/10.26493/1855-3974.157.fc1

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications