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[{"key": "dc.contributor.advisor", "value": "V\u00e4h\u00e4kangas, Antti", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Rajala, Kai", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Johansson, Jenna", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2018-11-23T13:27:00Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2018-11-23T13:27:00Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2018", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/60315", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa luokitellaan \u00e4\u00e4relliset ryhm\u00e4t isomorfiaa vaille kertalukuun $15$ asti. Lis\u00e4ksi tutkielma tarjoaa menetelmi\u00e4, joita soveltamalla \u00e4\u00e4rellisten ryhmien luokittelua olisi mahdollista jatkaa my\u00f6s suurempien kertalukujen tapauksessa. \u00c4\u00e4rellisten ryhmien luokittelussa keski\u00f6\u00f6n nousevat Sylowin lauseet, joiden avulla voidaan analysoida \u00e4\u00e4rellisten ryhmien rakenteita.\n\nLagrangen lauseen mukaan \u00e4\u00e4rellisen ryhm\u00e4n aliryhm\u00e4n kertaluku jakaa ryhm\u00e4n kertaluvun. Sen k\u00e4\u00e4nteinen tulos ei yleisesti p\u00e4de, mutta Sylowin ensimm\u00e4isess\u00e4 lauseessa k\u00e4\u00e4nteinen saadaan p\u00e4tem\u00e4\u00e4n sellaisille alkuluvun $p$ potensseille $p^k$, jotka jakavat ryhm\u00e4n kertaluvun. T\u00e4ll\u00f6in on siis olemassa \u00e4\u00e4rellisen ryhm\u00e4n aliryhm\u00e4, jonka kertaluku on $p^k$. Jos t\u00e4m\u00e4 kertaluku $p^k$ on suurin sellainen alkuluvun $p$ potenssi, joka jakaa ryhm\u00e4n kertaluvun, aliryhm\u00e4\u00e4 sanotaan Sylowin $p$-aliryhm\u00e4ksi. T\u00e4ll\u00f6in voidaan muotoilla Sylowin toinen lause, jonka mukaan \u00e4\u00e4rellisen ryhm\u00e4n Sylowin $p$-aliryhm\u00e4t konjugoivat kesken\u00e4\u00e4n. Edelleen voidaan osoittaa, ett\u00e4 t\u00e4llaiset Sylowin $p$-aliryhm\u00e4t ovat kesken\u00e4\u00e4n isomorfisia. Sylowin kolmas lause antaa puolestaan ehtoja n\u00e4iden Sylowin $p$-aliryhmien lukum\u00e4\u00e4r\u00e4lle. Sen mukaan Sylowin $p$-aliryhmien lukum\u00e4\u00e4r\u00e4 jakaa ryhm\u00e4n kertaluvun ja voidaan kirjoittaa muodossa $1+pk$ jollekin $k=0, 1, 2, \\ldots$.\n\nTutkielmassa luokitellaan ensin kaikki \u00e4\u00e4relliset Abelin ryhm\u00e4t. Kyseiset ryhm\u00e4t voidaan luokitella ilman Sylowin lauseiden apua, mutta niiden rakenteet noudattavat kuitenkin Sylowin lauseita. T\u00e4m\u00e4n j\u00e4lkeen siirryt\u00e4\u00e4n Sylowin lauseisiin, joita soveltaen p\u00e4\u00e4dyt\u00e4\u00e4n luokittelemaan yleisesti \u00e4\u00e4rellisi\u00e4 ryhmi\u00e4. Lopuksi kootaan ja viimeistell\u00e4\u00e4n \u00e4\u00e4rellisten ryhmien luokittelu isomorfiaa vaille kertalukuun $15$ asti.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2018-11-23T13:27:00Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2018-11-23T13:27:00Z (GMT). No. of bitstreams: 0\n Previous issue date: 2018", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "54", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Sylowin lauseet", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "\u00e4\u00e4rellisten ryhmien luokittelu", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Sylowin lauseet \u00e4\u00e4rellisten ryhmien luokittelussa", "language": "", "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-201811234865", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "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": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "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.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ryhm\u00e4teoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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