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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Krull–Schmidt theorem ： ウィキペディア英語版 Krull–Schmidt theorem In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups. ==Definitions== We say that a group ''G'' satisfies the ascending chain condition (ACC) on subgroups if every sequence of subgroups of ''G'': :$1\; =\; G\_0\; \backslash le\; G\_1\; \backslash le\; G\_2\; \backslash le\; \backslash cdots\backslash ,$ is eventually constant, i.e., there exists ''N'' such that ''G''''N'' = ''G''''N''+1 = ''G''''N''+2 = ... . We say that ''G'' satisfies the ACC on normal subgroups if every such sequence of normal subgroups of ''G'' eventually becomes constant. Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasing sequences of (normal) subgroups: :$G\; =\; G\_0\; \backslash ge\; G\_1\; \backslash ge\; G\_2\; \backslash ge\; \backslash cdots.\backslash ,$ Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group $\backslash mathbf$ satisfies ACC but not DCC, since (2) > (2)2 > (2)3 > ... is an infinite decreasing sequence of subgroups. On the other hand, the $p^\backslash infty$-torsion part of $\backslash mathbf/\backslash mathbf$ (the quasicyclic ''p''-group) satisfies DCC but not ACC. We say a group ''G'' is indecomposable if it cannot be written as a direct product of non-trivial subgroups ''G'' = ''H'' × ''K''. 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「Krull–Schmidt theorem」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.