20.500.12556/RUP-2751
Decomposition of skew-morphisms of cyclic groups
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 ▫$(n, \varphi(n)) = 1$▫.
cyclic group
permutation group
skew-morphism
Schur ring
ciklična grupa
permutacijska grupa
poševni morfizem
Schurov kolobar
true
true
false
Angleški jezik
Angleški jezik
Delo ni kategorizirano
2013-10-15 12:07:37
2013-10-15 12:07:37
2024-03-01 12:08:51
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2011
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str. 329-349
no. 2
Vol. 4
2011
0000-00-00
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1855-3966
519.1
1024369492
http://amc.imfm.si/index.php/amc/article/view/157/139
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https://repozitorij.upr.si/Dokument.php?lang=slv&id=2751
Inštitut Andrej Marušič
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