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Statements

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dbr:Emmy_Noether
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dbr:Fitting's_theorem
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yago:Message106598915 yago:Theorem106752293 yago:Communication100033020 yago:Statement106722453 yago:WikicatTheoremsInGroupTheory yago:Abstraction100002137 yago:WikicatTheoremsInAlgebra yago:Proposition106750804
rdfs:label
Fitting's theorem Satz von Fitting 菲廷定理 Teorema de Fitting
rdfs:comment
Fitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows: If M and N are nilpotent normal subgroups of a group G, then their product MN is also a nilpotent normal subgroup of G; if, moreover, M is nilpotent of class m and N is nilpotent of class n, then MN is nilpotent of class at most m + n. Teorema de Fitting es un teorema matemático demostrado por Hans Fitting. Se puede establecer de la siguiente manera: Si M y N son un subgrupo normal nilpotente de un grupo G, entonces su producto MN es también un subgrupo normal nilpotente de G; Si, además, M es nilpotente de clase m y N es nilpotente de clase n, entonces MN es nilpotente de clase a lo sumo m + n. 菲廷（德語：Fitting）定理是群論中冪零子群的一條定理，由證明。定理敘述如下： 若M和N是群G的冪零正規子群，則其乘積MN也是G的冪零正規子群。若M是冪零類m，N是冪零類n，則MN冪零類不大於m+n。 因此可知有限多個冪零正規子群生成的子群也是冪零群。這結果可以證明某些類的群（包括所有有限群）的是冪零群。但是無限多個冪零子群生成的子群不一定是冪零群。 用序理論的詞彙來說，菲廷定理的一部份可以表達為冪零正規子群族是一個。
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Teorema de Fitting es un teorema matemático demostrado por Hans Fitting. Se puede establecer de la siguiente manera: Si M y N son un subgrupo normal nilpotente de un grupo G, entonces su producto MN es también un subgrupo normal nilpotente de G; Si, además, M es nilpotente de clase m y N es nilpotente de clase n, entonces MN es nilpotente de clase a lo sumo m + n. Por inducción se deduce también que el subgrupo generado por una colección finita de subgrupos normales nilpotentes es nilpotente. Sin embargo, un subgrupo generado por una colección infinita de subgrupos normales nilpotentes no tiene que ser nilpotente. 菲廷（德語：Fitting）定理是群論中冪零子群的一條定理，由證明。定理敘述如下： 若M和N是群G的冪零正規子群，則其乘積MN也是G的冪零正規子群。若M是冪零類m，N是冪零類n，則MN冪零類不大於m+n。 因此可知有限多個冪零正規子群生成的子群也是冪零群。這結果可以證明某些類的群（包括所有有限群）的是冪零群。但是無限多個冪零子群生成的子群不一定是冪零群。 用序理論的詞彙來說，菲廷定理的一部份可以表達為冪零正規子群族是一個。 Fitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows: If M and N are nilpotent normal subgroups of a group G, then their product MN is also a nilpotent normal subgroup of G; if, moreover, M is nilpotent of class m and N is nilpotent of class n, then MN is nilpotent of class at most m + n. By induction it follows also that the subgroup generated by a finite collection of nilpotent normal subgroups is nilpotent. This can be used to show that the Fitting subgroup of certain types of groups (including all finite groups) is nilpotent. However, a subgroup generated by an infinite collection of nilpotent normal subgroups need not be nilpotent.
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