Decomposition of skew-morphisms of cyclic groups

Kovács, István; Nedela, Roman

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Naslov:	Decomposition of skew-morphisms of cyclic groups
Avtorji:	ID Kovács, István (Avtor) ID Nedela, Roman (Avtor)
Datoteke:	http://amc.imfm.si/index.php/amc/article/view/157/139
Jezik:	Angleški jezik
Vrsta gradiva:	Delo ni kategorizirano
Tipologija:	1.01 - Izvirni znanstveni članek
Organizacija:	IAM - Inštitut Andrej Marušič
Opis:	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 $(n, \varphi(n)) = 1$ .
Ključne besede:	cyclic group, permutation group, skew-morphism, Schur ring
Leto izida:	2011
Št. strani:	str. 329-349
Številčenje:	Vol. 4, no. 2
PID:	20.500.12556/RUP-2751
ISSN:	1855-3966
UDK:	519.1
COBISS.SI-ID:	1024369492
Datum objave v RUP:	15.10.2013
Število ogledov:	4901
Število prenosov:	112
Metapodatki:
:	KOVÁCS, István in NEDELA, Roman, 2011, Decomposition of skew-morphisms of cyclic groups. [na spletu]. 2011. Vol. 4, no. 2, p. 329–349. [Dostopano 3 april 2025]. Pridobljeno s: http://amc.imfm.si/index.php/amc/article/view/157/139 Kopiraj citat