Decomposition of skew-morphisms of cyclic groups

István Kovács (Author), Roman Nedela (Author)

Abstract

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 ▫$(n, \varphi(n)) = 1$▫.

Keywords

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

Data

Language:	English
Year of publishing:	2011
Typology:	1.01 - Original Scientific Article
Organization:	UP - University of Primorska
UDC:	519.1
COBISS:	1024369492
ISSN:	1855-3966
Parent publication:	Ars mathematica contemporanea
Views:	3667
Downloads:	108
Average score:	0 (0 votes)
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Other data

Secondary language:	English
Secondary keywords:	ciklična grupa;permutacijska grupa;poševni morfizem;Schurov kolobar;
Type (COBISS):	Not categorized
Pages:	str. 329-349
Volume:	ǂVol. ǂ4
Issue:	ǂno. ǂ2
Chronology:	2011
ID:	1477169