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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 Frobenius determinant theorem ： ウィキペディア英語版 Frobenius determinant theorem In mathematics, the Frobenius determinant theorem is a discovery made in 1896 by the mathematician Richard Dedekind, who wrote a letter to F. G. Frobenius about it (reproduced in , with an English translation in ). If one takes the multiplication table of a group ''G'' and replaces each entry ''g'' with the variable ''x''''g'', and subsequently takes the determinant, then the determinant factors as a product of ''n'' irreducible polynomials, where ''n'' is the number of conjugacy classes. Moreover, each polynomial is raised to a power equal to its degree. Frobenius proved this surprising fact, and this theorem became known as the Frobenius determinant theorem. ==Formal statement== Let a finite group $G$ have elements $g\_1,\; g\_2,\backslash dots,g\_n$, and let $x\_$ be associated with each element of $G$. Define the matrix $X\_G$ with entries $a\_=x\_$. Then : $\backslash det\; X\_G\; =\; \backslash prod\_^r\; P\_j(x\_,x\_,\backslash dots,x\_)^$ where ''r'' is the number of conjugacy classes of ''G''. 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「Frobenius determinant 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.