378 KB37 KB200820252022May, Coy L.Zimmerman, Jayštevilka:1letnik:22P1.09 (13 str.)DOI:10.26493/1855-3974.2257.6deCOBISSID_HOST:115590659ISSN:1855-3966URN:URN:NBN:SI:doc-UJWV8CKXenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneadelovanje grupegenusgroup actionmočni simetrični rodNEC groupNEC grupaRiemann surfaceRiemannova ploskevrodstrong symmetric genusMaximal order group actions on Riemann surfaces|A natural problem is to determine, for each value of the integer ?$g \ge 2$?, the largest order of a group that acts on a Riemann surface of genus ?$g$?. Let ?$N(g)$? (respectively ?$M(g)$?) be the largest order of a group of automorphisms of a Riemann surface of genus ?$g \ge 2$? preserving the orientation (respectively possibly reversing the orientation) of the surface. The basic inequalities comparing ?$N(g)$? and ?$M(g)$? are ?$N(g) \le M(g) \le 2N(g)$?. There are well-known families of extended Hurwitz groups that provide an infinite number of integers ?$g$? satisfying ?$M(g) = 2N(g)$?. It is also easy to see that there are solvable groups which provide an infinite number of such examples. We prove that, perhaps surprisingly, there are an infinite number of integers ?$g$? such that ?$N(g) = M(g)$?. Specifically, if ?$p$? is a prime satisfying ?$p \equiv 1 (mod 6)$? and ?$g = 3p + 1$? or ?$g = 2p+1$?, there is a group of order ?$24(g - 1)$? that acts on a surface of genus ?$g$? preserving the orientation of the surface. For all such values of $g$ larger than a fixed constant, there are no groups with order larger than ?$24(g - 1)$? that act on a surface of genus ?$g$?Kako določiti, za vsako vrednost celega števila ?$g \ge 2$?, največji red grupe, ki deluje na Riemannovi ploskvi rodu ?$g$?, je zelo naraven problem. Naj bo ?$N(g)$? največji red grupe avtomorfizmov Riemannove ploskve rodu ?$g \ge 2$?, ki ohranjajo orientacijo ploskve, ?$M(g)$? pa tistih, ki orientacijo morda obrnejo. Osnovne neenakosti, ki primerjajo ?$N(g)$? in ?$M(g)$?, so: ?$N(g) \le M(g) \le 2N(g)$?. Dobro znane so družine razširjenih Hurwitzevih grup, iz katerih dobimo neskončno mnogo celih števil ?$g$?, ki zadoščajo enakosti ?$M(g) = 2N(g)$?. Lahko je tudi videti, da obstajajo rešljive grupe, iz katerih dobimo neskončno mnogo takih primerov. Dokažemo, kar morda preseneča, da obstaja neskončno mnogo celih števil ?$g$?, za katera je ?$N(g) = M(g)$?. V primeru, da je ?$p$? praštevilo, ki zadošča ?$p \equiv 1 (mod 6)$? in ?$g = 3p + 1$? ali ?$g = 2p+1$?, obstaja grupa reda ?$24(g - 1)$?, ki deluje na neki ploskvi rodu ?$g$?, pri čemer ohranja njeno orientacijo. Za vse vrednosti ?$g$?, ki so večje od neke fiksne konstante, ne obstajajo grupe z redom večjim od ?$24(g - 1)$?, ki bi delovale na ploskvi rodu ?$g$?TEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije