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In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel.〔Jacobson (2009), p. 41〕 The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules and vector spaces are developed. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood. On the other hand, the theory of infinite abelian groups is an area of current research. == Definition == An abelian group is a set, ''A'', together with an operation • that combines any two elements ''a'' and ''b'' to form another element denoted . The symbol • is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, , must satisfy five requirements known as the ''abelian group axioms'': ;Closure: For all ''a'', ''b'' in ''A'', the result of the operation is also in ''A''. ;Associativity: For all ''a'', ''b'' and ''c'' in ''A'', the equation holds. ;Identity element: There exists an element ''e'' in ''A'', such that for all elements ''a'' in ''A'', the equation holds. ;Inverse element: For each ''a'' in ''A'', there exists an element ''b'' in ''A'' such that , where ''e'' is the identity element. ;Commutativity: For all ''a'', ''b'' in ''A'', ''a'' • ''b'' = ''b'' • ''a''. More compactly, an abelian group is a commutative group. A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group". 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「abelian group」の詳細全文を読む スポンサード リンク
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