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In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup ''H'' of a group ''G'' is normal in ''G'' if and only if ''gH'' = ''Hg'' for all ''g'' in ''G'', i.e., the sets of left and right cosets coincide. Normal subgroups (and ''only'' normal subgroups) can be used to construct quotient groups from a given group. Évariste Galois was the first to realize the importance of the existence of normal subgroups.〔 C.D. Cantrell, ''Modern Mathematical Methods for Physicists and Engineers.'' Cambridge University Press, 200, p 160.〕 == Definitions == A subgroup ''N'' of a group ''G'' is called a normal subgroup if it is invariant under conjugation; that is, for each element ''n'' in ''N'' and each ''g'' in ''G'', the element ''gng''−1 is still in ''N''.〔 〕 We write : For any subgroup, the following conditions are equivalent to normality. Therefore any one of them may be taken as the definition: *For all ''g'' in ''G'', ''gNg''−1 ⊆ ''N''. *For all ''g'' in ''G'', ''gNg''−1 = ''N''. *''g'',''h'' ∈ ''G'', ''gh'' ∈ ''N'' → ''hg'' ∈ ''N''. *The sets of left and right cosets of ''N'' in ''G'' coincide. *For all ''g'' in ''G'', ''gN'' = ''Ng''. *''N'' is a union of conjugacy classes of ''G''. *There is some homomorphism on ''G'' for which ''N'' is the kernel. The last condition accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images,〔 Pál Dõmõsi and Chrystopher L. Nehaniv, ''Algebraic Theory of Automata Networks (SIAM Monographs on Discrete Mathematics and Applications, 11)'', SIAM, 2004, p.7 〕 a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup. 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「normal subgroup」の詳細全文を読む スポンサード リンク
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