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In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: an ''inner'' semidirect product is a particular way in which a group can be constructed from two subgroups, one of which is a normal subgroup, while an ''outer'' semidirect product is a cartesian product as a set, but with a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as ''semidirect products''. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (aka split() extension). == Some equivalent definitions of inner semidirect products == Let ''G'' be a group with identity element ''e'', a subgroup ''H'' and a normal subgroup ''N'' (i.e., ). With this premise, the following statements are equivalent: * ''G'' = ''NH'' and ''N'' ∩ ''H'' = . * Every element of ''G'' can be written in a unique way as a product ''nh'', with and . * Every element of ''G'' can be written in a unique way as a product ''hn'', with and . * The natural embedding , composed with the natural projection , yields an isomorphism between ''H'' and the quotient group . * There exists a homomorphism that is the identity on ''H'' and whose kernel is ''N''. If one (and therefore all) of these statements hold, we say that ''G'' is a semidirect product of ''N'' and ''H'', written : or that ''G'' ''splits'' over ''N''; one also says that ''G'' is a semidirect product of ''H'' acting on ''N'', or even a semidirect product of ''H'' and ''N''. To avoid ambiguity, it is advisable to specify which of the two subgroups is normal. 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「Semidirect product」の詳細全文を読む スポンサード リンク
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