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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 Lagrange's theorem (group theory) ： ウィキペディア英語版 Lagrange's theorem (group theory) Lagrange's theorem, in the mathematics of group theory, states that for any finite group ''G'', the order (number of elements) of every subgroup ''H'' of ''G'' divides the order of ''G''. The theorem is named after Joseph-Louis Lagrange. == Proof of Lagrange's Theorem == This can be shown using the concept of left cosets of ''H'' in ''G''. The left cosets are the equivalence classes of a certain equivalence relation on ''G'' and therefore form a partition of ''G''. Specifically, ''x'' and ''y'' in ''G'' are related if and only if there exists ''h'' in ''H'' such that ''x = yh''. If we can show that all cosets of ''H'' have the same number of elements, then each coset of ''H'' has precisely |''H''| elements. We are then done since the order of ''H'' times the number of cosets is equal to the number of elements in ''G'', thereby proving that the order of ''H'' divides the order of ''G''. Now, if ''aH'' and ''bH'' are two left cosets of ''H'', we can define a map ''f'' : ''aH'' → ''bH'' by setting ''f''(''x'') = ''ba''−1''x''. This map is bijective because its inverse is given by $f^(y)\; =\; ab^y\backslash mbox$ This proof also shows that the quotient of the orders |''G''| / |''H''| is equal to the index (: ''H'' ) (the number of left cosets of ''H'' in ''G''). If we allow ''G'' and ''H'' to be infinite, and write this statement as :$\backslash left|G\backslash right|\; =\; \backslash left(:\; H\backslash right)\; \backslash cdot\; \backslash left|H\backslash right|\backslash mbox$ then, seen as a statement about cardinal numbers, it is equivalent to the Axiom of choice. 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「Lagrange's theorem (group theory)」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.