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 1 - 3 / 31 1.Decomposition of skew-morphisms of cyclic groupsIstván Kovács, Roman Nedela, 2011, original scientific articleAbstract: 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 besedahSummary of found: ...A skew-morphism of a group ▫$H$▫ is a permutation ▫$\sigma$▫ of its...Keywords: cyclic group, permutation group, skew-morphism, Schur ringPublished: 15.10.2013; Views: 1932; Downloads: 54 Full text (0,00 KB) 2.A note on finite non-solvable groups with few non-cyclic subgroupsJiangtao Shi, Cui Zhang, 2011, original scientific articleFound in: ključnih besedahSummary of found: ...finite group, non-cyclic subgroup, non-solvable group, ...Keywords: finite group, non-cyclic subgroup, non-solvable groupPublished: 15.10.2013; Views: 1195; Downloads: 18 Full text (0,00 KB) 3.Cyclic groups are CI-groups for balanced configurationsMikhail Muzychuk, Dragan Marušič, István Kovács, Sergio Hiroki Koike Quintanar, 2018, original scientific articleFound in: ključnih besedahSummary of found: ...Cayley object, CI-property, cyclic group, balanced configuration...Keywords: Cayley object, CI-property, cyclic group, balanced configurationPublished: 20.07.2018; Views: 991; Downloads: 163 Full text (0,00 KB)This document has more files! More...
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