326 KB36 KB170 KB4 KB170 KB4 KB200820252023O'Reilly-Regueiro, Eugenia O'Reilly-RegueiroRodríguez-Fitta, José Emanuelštevilka:2letnik:23P2.04 (14 str.)DOI:10.26493/1855-3974.2415.fd1COBISSID_HOST:150274307ISSN:1855-3966URN:URN:NBN:SI:doc-DKC631PCenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaautomorphism groups of designsflag-transitivitygrupa avtomorfizmov dizajnovpraporna tranzitivnostprimitive groupsprimitivne grupesimetrični dizajnisymmetric-designsA parametrisation for symmetric designs admitting a flag-transitive, point-primitive automorphism group with a product action|We study ?$(v, \, k,\, \lambda)$?-symmetric designs having a flag-transitive, point-primitive automorphism group, with ?$v=m^2$? and ?$(k, \, \lambda)=t>1$?, and prove that if ?$D$? is such a design with ?$m$? even admitting a flag-transitive, point-primitive automorphism group ?$G$?, then either: (1) ?$D$? is a design with parameters ?$\left((2t+s-1)^2, \frac{2t^2-(2-s)t)}{s}, \frac{t^2-t)}{s^2}\right)$? with ?$s\ge 1$? odd, or (2) ?$G$? does not have a non-trivial product action. We observe that the parameters in (1), when ?$s=1$?, correspond to Menon designs. We also prove that if ?$D$? is a ?$(v, \, k,\, \lambda)$?-symmetric design with a flag-transitive, point-primitive automorphism group of product action type with ?$v=m^l$? and ?$l\ge 2$? then the complement of ?$D$? does not admit a flag-transitive automorphism groupPreučujemo ?$(v, \, k,\, \lambda)$?-simetrične dizajne, ki imajo praporno tranzitivno in točkovno primitivno grupo avtomorfizmov, kjer je ?$v=m^2$? in ?$(k, \, \lambda)=t>1$?, ter dokažemo, da če je ?$D$? dizajn s sodim ?$m$?, ki dopušča praporno tranzitivno in točkovno primitivno grupo avtomorfizmov ?$G$?, potem velja bodisi: (1) ?$D$? je dizajn s parametri ?$\left((2t+s-1)^2, \frac{2t^2-(2-s)t)}{s}, \frac{t^2-t)}{s^2}\right)$?, kjer je ?$s\ge 1$? lih, bodisi (2) ?$G$? nima netrivialnega produktnega delovanja. Opazimo, da parametri v (1), kadar je ?$s=1$?, ustrezajo Menonovim dizajnom. Dokažemo tudi, da če je ?$D$ $(v, \, k,\, \lambda)$?-simetričen dizajn s praporno tranzitivno in točkovno primitivno grupo avtomorfizmov s produktnim delovanjem, kjer je ?$v=m^l$? in ?$l\ge 2$?, potem komplement dizajna ?$D$? ne dopušča praporno tranzitivne grupe avtomorfizovTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije