翻訳と辞書 Words near each other ・ Group Voyagers ・ Group Hug Tour ・ Group I catalytic intron ・ Group I pyridoxal-dependent decarboxylases ・ Group identifier ・ Group II intron ・ Group II pyridoxal-dependent decarboxylases ・ Group III ・ Group III intron ・ Group III pyridoxal-dependent decarboxylases ・ Group information management ・ Group inou ・ Group insurance ・ Group intelligence ・ Group inverse ・ Group isomorphism ・ Group isomorphism problem ・ Group IV pyridoxal-dependent decarboxylases ・ Group JK corynebacterium sepsis ・ Group Joos ・ Group key ・ Group Kyushu ・ Group Litigation Order ・ Group Marriage ・ Group marriage ・ Group Medal of the Four Day Marches ・ Group medical practice in the United States ・ Group method of data handling ・ Group mind (science fiction) ・ Group MOST 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 Group isomorphism ： ウィキペディア英語版 Group isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. == Definition and notation== Given two groups (G, ∗) and (H, $\backslash odot$), a ''group isomorphism'' from (G, ∗) to (H, $\backslash odot$) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function $f\; :\; G\; \backslash rightarrow\; H$ such that for all u and v in G it holds that : $f(u$ * v) = f(u) \odot f(v). The two groups (G, ∗) and (H, $\backslash odot$) are isomorphic if there exists an isomorphism from one to the other. This is written: : $(G,$ *) \cong (H, \odot) Often shorter and simpler notations can be used. When the relevant group operations are unambiguous they are omitted and one writes: : $G\; \backslash cong\; H$ Sometimes one can even simply write G = H. Whether such a notation is possible without confusion or ambiguity depends on context. For example, the equals sign is not very suitable when the groups are both subgroups of the same group. See also the examples. Conversely, given a group (G, ∗), a set H, and a bijection $f\; :\; G\; \backslash rightarrow\; H$, we can make H a group (H, $\backslash odot$) by defining : $f(u)\; \backslash odot\; f(v)\; =\; f(u$ * v). If H = G and $\backslash odot$ = ∗ then the bijection is an automorphism (''q.v.''). Intuitively, group theorists view two isomorphic groups as follows: For every element ''g'' of a group ''G'', there exists an element ''h'' of ''H'' such that ''h'' 'behaves in the same way' as ''g'' (operates with other elements of the group in the same way as ''g''). For instance, if ''g'' generates ''G'', then so does ''h''. This implies in particular that ''G'' and ''H'' are in bijective correspondence. Thus, the definition of an isomorphism is quite natural. An isomorphism of groups may equivalently be defined as an invertible morphism in the category of groups, where invertible here means has a two-sided inverse. 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「Group isomorphism」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.