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In algebra, a cyclic group is a group that is generated by a single element. That is, it consists of a set of elements with a single invertible associative operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation or its inverse to ''g''. Each element can be written as a power of ''g'' in multiplicative notation, or as a multiple of ''g'' in additive notation. This element ''g'' is called a ''generator'' of the group.〔 Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups. ==Definition== A group ''G'' is called cyclic if there exists an element ''g'' in ''G'' such that Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group ''G'' that contains ''g'' is ''G'' itself suffices to show that ''G'' is cyclic. For example, if ''G'' = is a group, then ''g''6 = ''g''0, and ''G'' is cyclic. In fact, ''G'' is essentially the same as (that is, isomorphic to) the set with addition modulo 6. For example, corresponds to , and corresponds to , and so on. One can use the isomorphism χ defined by . The name "cyclic" may be misleading:〔.〕 it is possible to generate infinitely many elements and not form any literal cycles; that is, every ''g''''n'' is distinct. (It can be thought of as having one infinitely long cycle.) A group generated in this way (for example, the first frieze group, p1) is called an infinite cyclic group, and is isomorphic to the additive group of the integers, . 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「cyclic group」の詳細全文を読む スポンサード リンク
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