|
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements ''x''1, ..., ''x''''s'' in ''G'' such that every ''x'' in ''G'' can be written in the form :''x'' = ''n''1''x''1 + ''n''2''x''2 + ... + ''n''''s''''x''''s'' with integers ''n''1, ..., ''n''''s''. In this case, we say that the set is a ''generating set'' of ''G'' or that ''x''1, ..., ''x''''s'' ''generate'' ''G''. Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below. ==Examples== * The integers are a finitely generated abelian group. * The integers modulo , are a finitely generated abelian group. * Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group. * Every lattice forms a finitely generated free abelian group. There are no other examples (up to isomorphism). In particular, the group of rational numbers is not finitely generated:〔Silverman & Tate (1992), (p. 102 )〕 if are rational numbers, pick a natural number coprime to all the denominators; then cannot be generated by . The group of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition and real numbers under multiplication are also not finitely generated.〔〔La Harpe (2000), (p. 46 )〕 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「Finitely generated abelian group」の詳細全文を読む スポンサード リンク
|