Complete regular dessins and skew-morphisms of cyclic groups

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Complete regular dessins and skew-morphisms of cyclic groups

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Complete regular dessins and skew-morphisms of cyclic groups

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/LO1506" target="_blank" >LO1506: Podpora udržitelnosti centra NTIS - Nové technologie pro informační společnost</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

ARS MATHEMATICA CONTEMPORANEA

ISSN

1855-3966

e-ISSN

—

Svazek periodika

18

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

SI - Slovinská republika

Počet stran výsledku

19

Strana od-do

289-307

Kód UT WoS článku

000581926200007

EID výsledku v databázi Scopus

2-s2.0-85095597179