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[{"key": "dc.contributor.advisor", "value": "Rajala, Kai", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Hautala, Sessi", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-05-27T07:00:13Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-05-27T07:00:13Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/102783", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tarkastella ryhm\u00e4teorian k\u00e4sitteit\u00e4 ja\nsoveltaa niit\u00e4 kuution v\u00e4ritysongelman ratkaisemiseksi. Ryhm\u00e4teoriassa tarkastellaan symmetrioita sek\u00e4 kiertoja ja peilauksia. Platonin kappaleista k\u00e4sitell\u00e4\u00e4n erityisesti kuutiota. Lis\u00e4ksi todistetaan erin\u00e4isi\u00e4 lauseita, jotka kertovat yhdisteen symmetriasta ja kierroista.\nTutkielman toinen keskeinen aihe on ryhmien toiminnat ja siihen liittyv\u00e4t k\u00e4sitteet rata ja stabilisaattori. N\u00e4ist\u00e4 annetaan muutamia esimerkkej\u00e4 erityisesti kuution tapauksessa. Lis\u00e4ksi todistetaan, ett\u00e4 stabilisaattori\non aliryhm\u00e4. T\u00e4rke\u00e4n\u00e4 lauseena on radan ja stabilisaattorin yhdist\u00e4v\u00e4 ratastabilisaattori-lause, joka yhdist\u00e4\u00e4 sivuluokat, radan ja stabilisaattorin.\nKonjugaattiluokkien avulla voidaan ryhm\u00e4n alkiot ryhmitell\u00e4 siten, ett\u00e4\nne k\u00e4ytt\u00e4ytyv\u00e4t samalla tavalla toiminnan suhteen. Konjugaattiluokat mahdollistavat tutkielman toisen t\u00e4rke\u00e4n aiheen, kuution v\u00e4ritysongelmien, laskemisen matemaattisesti.\nToinen tutkielman p\u00e4\u00e4tulos on kuution v\u00e4ritysongelman tutkiminen ja\nmiten kuution kiertoryhm\u00e4\u00e4n vaikuttaa toiminta v\u00e4ritysten joukossa. T\u00e4h\u00e4n\ntarvitaan ratojen laskus\u00e4\u00e4nt\u00f6\u00e4 eli Burnsiden lemmaa. Sen avulla yhdistet\u00e4\u00e4n konjugaattiluokat ja kierrot kuution v\u00e4ritysongelmaan. Tavoitteena on\nselvitt\u00e4\u00e4, miten v\u00e4rien m\u00e4\u00e4r\u00e4n ja tahkojen jako vaikuttavat mahdollisiin v\u00e4rityksiin. Tutkielmassa my\u00f6s vertaillaan, miten v\u00e4rien m\u00e4\u00e4r\u00e4n lis\u00e4\u00e4minen ja\ntahkon jakaminen osiin vaikuttaa v\u00e4ritysmahdollisuuksien m\u00e4\u00e4r\u00e4\u00e4n.\nTutkielmassa on yhdistetty algebran teoreettista taustaa ja ryhm\u00e4teoriaa k\u00e4yt\u00e4nn\u00f6n sovelluksiin kuution ja geometrian avulla. Tutkielman avulla n\u00e4hd\u00e4\u00e4n, miten ryhm\u00e4teoriaa voidaan hy\u00f6dynt\u00e4\u00e4 k\u00e4yt\u00e4nn\u00f6ss\u00e4 ongelmien\nmallintamiseen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-05-27T07:00:13Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-05-27T07:00:13Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "27", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Ryhmien toiminnat ja kuution v\u00e4ritysongelma", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202505274608", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": "fi", "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ryhm\u00e4teoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ryhm\u00e4t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "symmetria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "radat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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