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In mathematics, a group ''G'' is called the direct sum 〔Homology. Saunders MacLane. Springer, Berlin; Academic Press, New York, 1963.〕〔László Fuchs. Infinite Abelian Groups〕 of two subgroups ''H''''1'' and ''H''''2'' if * each ''H''''1'' and ''H''''2'' are normal subgroups of ''G'' * the subgroups ''H''''1'' and ''H''''2'' have trivial intersection (i.e., having only the identity element in common), and * ''G'' = <''H''''1'', ''H''''2''>; in other words, ''G'' is generated by the subgroups ''H''''1'' and ''H''''2''. More generally, ''G'' is called the direct sum of a finite set of subgroups if * each ''H''''i'' is a normal subgroup of ''G'' * each ''H''''i'' has trivial intersection with the subgroup <>, and * ''G'' = <>; in other words, ''G'' is generated by the subgroups . If ''G'' is the direct sum of subgroups ''H'' and ''K'', then we write ''G'' = ''H'' + ''K''; if ''G'' is the direct sum of a set of subgroups , we often write ''G'' = ∑''H''''i''. Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups. In abstract algebra, this method of construction can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information. This notation is commutative; so that in the case of the direct sum of two subgroups, ''G'' = ''H'' + ''K'' = ''K'' + ''H''. It is also associative in the sense that if ''G'' = ''H'' + ''K'', and ''K'' = ''L'' + ''M'', then ''G'' = ''H'' + (''L'' + ''M'') = ''H'' + ''L'' + ''M''. A group which can be expressed as a direct sum of non-trivial subgroups is called ''decomposable''; otherwise it is called ''indecomposable''. If ''G'' = ''H'' + ''K'', then it can be proven that: * for all ''h'' in ''H'', ''k'' in ''K'', we have that ''h'' *''k'' = ''k'' *''h'' * for all ''g'' in ''G'', there exists unique ''h'' in ''H'', ''k'' in ''K'' such that ''g'' = ''h'' *''k'' * There is a cancellation of the sum in a quotient; so that (''H'' + ''K'')/''K'' is isomorphic to ''H'' The above assertions can be generalized to the case of ''G'' = ∑''H''''i'', where is a finite set of subgroups. * if ''i'' ≠ ''j'', then for all ''h''''i'' in ''H''''i'', ''h''''j'' in ''H''''j'', we have that ''h''''i'' * ''h''''j'' = ''h''''j'' * ''h''''i'' * for each ''g'' in ''G'', there unique set of such that :''g'' = ''h''1 *''h''2 * ... * ''h''''i'' * ... * ''h''''n'' * There is a cancellation of the sum in a quotient; so that ((∑''H''''i'') + ''K'')/''K'' is isomorphic to ∑''H''''i'' Note the similarity with the direct product, where each ''g'' can be expressed uniquely as :''g'' = (''h''1,''h''2, ..., ''h''''i'', ..., ''h''''n'') Since ''h''''i'' * ''h''''j'' = ''h''''j'' * ''h''''i'' for all ''i'' ≠ ''j'', it follows that multiplication of elements in a direct sum is isomorphic to multiplication of the corresponding elements in the direct product; thus for finite sets of subgroups, ∑''H''''i'' is isomorphic to the direct product ×. ==Direct summand== Given a group , we say that a subgroup is a direct summand of (or that splits form ) if and only if there exist another subgroup such that is the direct sum of the subgroups and In abelian groups, if is a divisible subgroup of then is a direct summand of . 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「Direct sum of groups」の詳細全文を読む スポンサード リンク
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