Title: Complete regular dessins and skew-morphisms of cyclic groups
Other Titles: Úplné regulární dezény a kosomorfismy cyklických grup
Authors: Feng, Yan-Quan
Hu, Kan
Nedela, Roman
Škoviera, Martin
Wang, Na-Er
Citation: FENG, Y., HU, K., NEDELA, R., ŠKOVIERA, M., WANG, N. Complete regular dessins and skew-morphisms of cyclic groups. Ars mathematica contemporanea, 2020, roč. 18, č. 2, s. 289-307. ISSN 1855-3966.
Issue Date: 2020
Publisher: Society of Mathematicians, Physicists and Astronomers of Slovenia
Document type: článek
URI: 2-s2.0-85095597179
ISSN: 1855-3966
Keywords: Grothendieckuv dezén, grupa automorfismů, kompletní bipartitní graf, kosomorfismy grup, cyklické grupy
Keywords in different language: Grothendieck dessin, automorphism group, complete bipartite graph, skew-morphism, cyclic group
Abstract: Grothendickov dezén je celulární dekompozice orientovatelné kompaktní souvislé plochy s bipartitním grafem s fixovaným 2-obarvením uzlů. V článku studujeme dezény, kterých graf je kompletní bipartitní graf Km,n a grupa automorfismů je tranzitivní na množině hran. V práci dokazujeme souvis kompletních regulárních (m,n)-dezénů s dvojicemi recipročných kosomorfizmů cyklických grup řádu m a n. Práce obsahuje klasifikaci dezénů v případě, že existuje jediný dezén pro dané m a n.
Abstract in different language: 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.
Rights: © Society of Mathematicians, Physicists and Astronomers of Slovenia
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