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Complete regular dessins and skew-morphisms of cyclic groupsFeng, Yan-Quan ...Risba je 2-celična vložitev povezanega 2-barvnega dvodelnega grafa na orientabilno sklenjeno ploskev. Risba je pravilna, če njena grupa avtomorfizmov, ki ohranjajo orientacijo in barve, deluje ... pravilno na povezavah. V tem članku preučujemo pravilne risbe, katerih osnovni graf je polni dvodelni graf ▫$K_{m,n}$▫, imenovane ▫$(m,n)$▫-polne pravilne risbe. Na ta način vzpostavimo precej presenetljivo korespondenco med ▫$(m,n)$▫-polnimi pravilnimi risbami in pari poševnih morfizmov cikličnih grup. Poševni morfizem končne grupe ▫$A$▫ je bijekcija ▫$\varphi\colon A\to A$▫, ki zadošča identiteti ▫$\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$▫ za neko funkcijo ▫$\pi\colon A\to\mathbb{Z}$▫ in fiksira nevtralni element grupe ▫$A$▫. Dokažemo, da vsaka ▫$(m,n)$▫-polna pravilna risba ▫$\mathcal{D}$▫ določa par recipročnih poševnih morfizmov cikličnih grup ▫$\mathbb{Z}_n$▫ in ▫$\mathbb{Z}_m$▫. Velja tudi obratno, ▫$\mathcal{D}$▫ lahko rekonstruiramo iz takšnega recipročnega para. Na podlagi tega dokažemo, da so polne pravilne risbe, eksaktne biciklične grupe z izbranim parom generatorjev, ter pari recipročnih poševnih morfizmov cikličnih grup vsi v povratno enolični korespondenci. Nazadnje pa uporabimo naš glavni rezultat še za določitev vseh parov celih števil ▫$m$▫ in ▫$n$▫, za katere obstaja, do zamenjave barv natančno, samo en izomorfnostni razred $(m,n)$-polnih regularnih risb. Dokažemo, da se to zgodi natanko takrat, ko je vsaka grupa, izrazljiva kot produkt cikličnih grup reda ▫$m$▫ in ▫$n$▫, abelska, kar se naposled prevede na pogoj ▫$\gcd(m,\phi(n))=\gcd(\phi(m),n)=1$▫, kjer je ▫$\phi$▫ Eulerjeva funkcija.Source: Ars mathematica contemporanea. - ISSN 1855-3966 (Vol. 18, no. 2, 2020, str. 289-307)Type of material - article, component part ; adult, seriousPublish date - 2020Language - englishCOBISS.SI-ID - 41168899
Author
Feng, Yan-Quan |
Hu, Kan |
Nedela, Roman |
Škoviera, Martin |
Wang, Naer
Topics
pravilna risba |
biciklična grupa |
poševni morfizem |
vložitev grafa |
regular dessin |
bicyclic group |
skew-morphism |
graph embedding
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Database name | Field | Year |
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Links to authors' personal bibliographies | Links to information on researchers in the SICRIS system |
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Feng, Yan-Quan | |
Hu, Kan | 57045 |
Nedela, Roman | 27776 |
Škoviera, Martin | |
Wang, Naer |
Source: Personal bibliographies
and: SICRIS
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