翻訳と辞書 Words near each other ・ Solway Junction Railway ・ Solway Lass ・ Solway Moss ・ Solway Plain ・ Solway Railway Station ・ Solway Sharks ・ Solway Star F.C. ・ Solway Township, St. Louis County, Minnesota ・ Solway, Minnesota ・ Solway, New Zealand ・ Solway, Tennessee ・ Solwezi ・ Solva Saal ・ Solva Sawan ・ Solvable ・ Solvable group ・ Solvable Lie algebra ・ Solvac ・ Solvace ・ Solvalla ・ Solvalla Stockholm ・ Solvalou ・ Solvang (company) ・ Solvang Formation ・ Solvang Vintage Motorcycle Museum ・ Solvang, California ・ Solvang, Vestfold ・ Solvarbo ・ Solvated electron ・ Solvation Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Solvable group ： ウィキペディア英語版 Solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable. ==Definition== A group $G$ is called solvable if it has a subnormal series whose factor groups are all abelian, that is, if there are subgroups such that $G\_$ is normal in $G\_j$, and $G\_j/G\_$ is an abelian group, for $j=1,2,\backslash dots,k$. Or equivalently, if its derived series, the descending normal series :$G\backslash triangleright\; G^\backslash triangleright\; G^\; \backslash triangleright\; \backslash cdots,$ where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup of ''G''. These two definitions are equivalent, since for every group ''H'' and every normal subgroup ''N'' of ''H'', the quotient ''H''/''N'' is abelian if and only if ''N'' includes ''H''(1). The least ''n'' such that $G^=\backslash$ is called the derived length of the solvable group ''G''. For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite group has finite composition length, and every simple abelian group is cyclic of prime order. The Jordan–Hölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to ''n''th roots (radicals) over some field. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series , with its only factor group isomorphic to Z, proves that it is in fact solvable. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Solvable group」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.