Decomposition of skew-morphisms of cyclic groups

István Kovács (Avtor), Roman Nedela (Avtor)

Povzetek

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

ciklična grupa;permutacijska grupa;poševni morfizem;Schurov kolobar;cyclic group;permutation group;skew-morphism;Schur ring;

Podatki

Jezik:	Angleški jezik
Leto izida:	2011
Tipologija:	1.01 - Izvirni znanstveni članek
Organizacija:	UP - Univerza na Primorskem
UDK:	519.1
COBISS:	1024369492
ISSN:	1855-3966
Matična publikacija:	Ars mathematica contemporanea
Št. ogledov:	3667
Št. prenosov:	108
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Sekundarni jezik:	Angleški jezik
Sekundarne ključne besede:	ciklična grupa;permutacijska grupa;poševni morfizem;Schurov kolobar;
Vrsta dela (COBISS):	Delo ni kategorizirano
Strani:	str. 329-349
Letnik:	ǂVol. ǂ4
Zvezek:	ǂno. ǂ2
Čas izdaje:	2011
ID:	1477169