VSE knjižnice (vzajemna bibliografsko-kataložna baza podatkov COBIB.SI)
  • Decomposition of skew-morphisms of cyclic groups
    Kovács, István, 1969- ; Nedela, Roman
    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 ▫$(n, \varphi(n)) = 1$▫.
    Vir: Ars mathematica contemporanea. - ISSN 1855-3966 (Vol. 4, no. 2, 2011, str. 329-349)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2011
    Jezik - angleški
    COBISS.SI-ID - 1024369492

    Povezava(-e):

    http://amc.imfm.si/index.php/amc/article/view/157/139
    Digitalna knjižnica Slovenije - dLib.si
    Repozitorij Univerze na Primorskem - RUP

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