Complete regular dessins and skew-morphisms of cyclic groups
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F20%3A43959631" target="_blank" >RIV/49777513:23520/20:43959631 - isvavai.cz</a>
Result on the web
<a href="https://amc-journal.eu/index.php/amc/article/view/1748" target="_blank" >https://amc-journal.eu/index.php/amc/article/view/1748</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.26493/1855-3974.1748.ebd" target="_blank" >10.26493/1855-3974.1748.ebd</a>
Alternative languages
Result language
angličtina
Original language name
Complete regular dessins and skew-morphisms of cyclic groups
Original language description
A dessin is a 2-cell embedding of a connected 2-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular dessins whose underlying graph is a complete bipartite graph Km;n, called (m; n)-complete regular dessins. The purpose is to establish a rather surprising correspondence between (m; n)- complete regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism of a finite group A is a permutation of A that satisfies the identity f(xy) = f(x)(f(y))^p(x) for some indeger valued function defined on A , moreover, f fixes the neutral element of A. We show that every (m; n)-complete regular dessin D determines a pair of reciprocal skew-morphisms of the cyclic groups Z_n and Z_m. Conversely, D can be reconstructed from such a reciprocal pair. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and pairs of reciprocal skew-morphisms of cyclic groups are all in a one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers m and n for which there exists, up to interchange of colours, exactly one isomorphism class of (m; n)-complete regular dessins. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order m and n is abelian, which eventually comes down to the condition gcd(m; e(n)) = gcd(e(m); n) = 1, where e is Euler’s totient function.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/LO1506" target="_blank" >LO1506: Sustainability support of the centre NTIS - New Technologies for the Information Society</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
ARS MATHEMATICA CONTEMPORANEA
ISSN
1855-3966
e-ISSN
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Volume of the periodical
18
Issue of the periodical within the volume
2
Country of publishing house
SI - SLOVENIA
Number of pages
19
Pages from-to
289-307
UT code for WoS article
000581926200007
EID of the result in the Scopus database
2-s2.0-85095597179