Decomposition of skew-morphisms of cyclic groups
Keywords:Cyclic group, permutation group, skew-morphism, Schur ring
AbstractA 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)σk(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 Zn: 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 Zn are obtained as σ = σ1 × σ2, where σi are skew-morphisms of Zn_i, i = 1, 2. As a consequence we obtain the following result: All skew-morphisms of Zn are automorphisms of Zn if and only if n = 4 or (n, φ(n)) = 1.
GEMS 2009 - Tale, Slovakia
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