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In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted . Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups. == Definition == Given groups and , the direct product is defined as follows: # The elements of are ordered pairs , where and . That is, the set of elements of is the Cartesian product of the sets and . # The binary operation on is defined componentwise: The resulting algebraic object satisfies the axioms for a group. Specifically: ;Associativity: The binary operation on is indeed associative. ;Identity: The direct product has an identity element, namely , where is the identity element of and is the identity element of . ;Inverses: The inverse of an element of is the pair , where is the inverse of in , and is the inverse of in . 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 product of groups」の詳細全文を読む スポンサード リンク
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