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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 Correspondence theorem (group theory) ： ウィキペディア英語版 Correspondence theorem (group theory) In the area of mathematics known as group theory, the correspondence theorem, sometimes referred to as the fourth isomorphism theorem〔〔Some authors use "fourth isomorphism theorem" to designate the Zassenhaus lemma; see for example by Alperin & Bell (p. 13) or 〕〔Depending how one counts the isomorphism theorems, the correspondence theorem can also be called the 3rd isomorphism theorem; see for instance H.E. Rose (2009), p. 78.〕 or the lattice theorem,〔W.R. Scott: ''Group Theory'', Prentice Hall, 1964, p. 27.〕 states that if $N$ is a normal subgroup of a group $G$, then there exists a bijection from the set of all subgroups $A$ of $G$ containing $N$, onto the set of all subgroups of the quotient group $G/N$. The structure of the subgroups of $G/N$ is exactly the same as the structure of the subgroups of $G$ containing $N,$ with $N$ collapsed to the identity element. This establishes a monotone Galois connection between the lattice of subgroups of $G$ and the lattice of subgroups of $G/N$, where the associated closure operator on subgroups of $G$ is $\backslash bar\; H\; =\; HN.$ Specifically, if :''G'' is a group, :''N'' is a normal subgroup of ''G'', :$\backslash mathcal$ is the set of all subgroups ''A'' of ''G'' such that $N\backslash subseteq\; A\backslash subseteq\; G$, and :$\backslash mathcal$ is the set of all subgroups of ''G/N'', then there is a bijective map $\backslash phi:\backslash mathcal\backslash to\backslash mathcal$ such that :$\backslash phi(A)=A/N$ for all $A\backslash in\; \backslash mathcal.$ One further has that if ''A'' and ''B'' are in $\backslash mathcal$, and ''A' = A/N'' and ''B' = B/N'', then *$A\; \backslash subseteq\; B$ if and only if $A\text{'}\; \backslash subseteq\; B\text{'}$; *if $A\; \backslash subseteq\; B$ then $|B:A|\; =\; |B\text{'}:A\text{'}|$, where $|B:A|$ is the index of ''A'' in ''B'' (the number of cosets ''bA'' of ''A'' in ''B''); *$\backslash langle\; A,B\backslash rangle\; /\; N\; =\; \backslash langle\; A\text{'},B\text{'}\; \backslash rangle,$ where $\backslash langle\; A,B\; \backslash rangle$ is the subgroup of $G$ generated by $A\backslash cup\; B;$ * $(A\backslash cap\; B)/N\; =\; A\text{'}\; \backslash cap\; B\text{'}$, and * $A$ is a normal subgroup of $G$ if and only if $A\text{'}$ is a normal subgroup of $G/N$. This list is far from exhaustive. In fact, most properties of subgroups are preserved in their images under the bijection onto subgroups of a quotient group. Similar results hold for rings, modules, vector spaces, and algebras. == See also == * Modular lattice 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「Correspondence 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.