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Decomposition of skew-morphisms of cyclic groups
István Kovács, Roman Nedela, 2011, original scientific article

Abstract: A skew-morphism of a group ▫$H$▫ is a permutation ▫$\sigma$▫ of its elements fixing the identity such that for every ▫$x, y \in H$▫ there exists an integer ▫$k$▫ such that ▫$\sigma (xy) = \sigma (x)\sigma k(y)$▫. 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 ▫$\mathbb Z_n$▫: if ▫$n = n_{1}n_{2}$▫ such that ▫$(n_{1}n_{2}) = 1$▫, and ▫$(n_{1}, \varphi (n_{2})) = (\varphi (n_{1}), n_{2}) = 1$▫ (▫$\varphi$▫ denotes Euler's function) then all skew-morphisms ▫$\sigma$▫ of ▫$\mathbb Z_n$▫ are obtained as ▫$\sigma = \sigma_1 \times \sigma_2$▫, where ▫$\sigma_i$▫ are skew-morphisms of ▫$\mathbb Z_{n_i}, \; i = 1, 2$▫. As a consequence we obtain the following result: All skew-morphisms of ▫$\mathbb Z_n$▫ are automorphisms of ▫$\mathbb Z_n$▫ if and only if ▫$n = 4$▫ or ▫$(n, \varphi(n)) = 1$▫.
Found in: ključnih besedah
Summary of found: ...groups in the context of the associated Schur rings. We prove the following decomposition theorem...
Keywords: cyclic group, permutation group, skew-morphism, Schur ring
Published: 15.10.2013; Views: 1935; Downloads: 55
URL Full text (0,00 KB)

CI-property for decomposable Schur rings over an Abelian group
István Kovács, Grigory Ryabov, 2019, original scientific article

Found in: ključnih besedah
Keywords: isomorphism, CI-group, Schur ring
Published: 17.12.2018; Views: 216; Downloads: 45
URL Full text (0,00 KB)

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