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Ryhmäteoria


triviaali ryhmänormaalit aliryhmättriviaali ryhmäAbelin ryhmäsyklinenkertalukualkulukualternoiva ryhmäPSL(2,7)alkuluvuistakokonaisluvutJordanin–Hölderin lauseenaFeitin–Thompsonin lauseenratkeavaYksinkertainen Lien ryhmäThompsonin ryhmätSchreierin otaksumanulkoisten automorfismienluokittelulauseen





Matematiikassa ryhmää G sanotaan yksinkertaiseksi jos se ei ole triviaali ryhmä ja sen ainoat normaalit aliryhmät ovat triviaali ryhmä e ja G.


Abelin ryhmä on yksinkertainen jos ja vain jos se on syklinen ja sen kertaluku on alkuluku. Ei-Abelin ryhmien luokittelu on hankalampaa. Pienin ei-Abelin yksinkertainen ryhmä on alternoiva ryhmä A5, jonka kertaluku on 60, ja toisaalta jokainen kertalukua 60 oleva yksinkertainen ryhmä on isomorfinen tämän kanssa. Toiseksi pienintä kertalukua oleva ei-Abelin ryhmä on kertalukua 168 ja se on isomorfinen PSL(2,7):n kanssa. Edelleen jokainen yksinkertainen kertalukua 168 oleva ryhmä on isomorfinen PSL(2,7):n kanssa.


Äärelliset yksinkertaiset ryhmät ovat tärkeitä, sillä ne ovat eräänlaisia äärellisten ryhmien "perusrakennuspalikoita", samoin kuin alkuluvuista voidaan rakentaa kaikki kokonaisluvut. Tämä tulos tunnetaan Jordanin–Hölderin lauseena. Matemaatikkojen suuri projekti, kaikkien äärellisten yksinkertaisten ryhmien luokittelu saatiin päätökseen vuonna 1982.


Feitin–Thompsonin lauseen mukaan jokainen parittoman kertaluvun ryhmä on ratkeava. Siten kaikilla äärellisillä yksinkertaisilla ryhmillä on parillinen kertaluku ellei ryhmä ole syklinen ja sen kertaluku alkuluku.


Äärettömän kertaluvun yksinkertaisia ryhmiä on myös olemassa. Yksinkertainen Lien ryhmä ja äärettömät Thompsonin ryhmät ovat esimerkkejä tällaisista.


Schreierin otaksuman mukaan jokainen äärellinen yksinkertainen ulkoisten automorfismien muodostama ryhmä on ratkeava. Tämä voidaan todistaa luokittelulauseen avulla.



Kirjallisuutta |


  • Häsä, Jokke; Rämö, Johanna: Johdatus abstraktiin algebraan. Helsinki: Gaudeamus, 2015. ISBN 978-952-495-361-0.

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