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In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order. That is, multiples of any element other than the identity element generate an infinite number of distinct elements of the group. == Definitions == (詳細はset ''G'', together with a binary operation * on ''G'', such that the following axioms are satisfied: ;Associativity: For all ''a'', ''b'' and ''c'' in ''G'', (''a'' * ''b'') *''c'' = ''a'' * (''b'' * ''c''). ;Identity element: There is an element ''e'' in ''G'', such that ''e'' * ''x'' = ''x'' * ''e'' = ''x'' for all ''x'' in ''G''. This element ''e'' is an identity element for * on ''G''. ;Inverse element: For each ''a'' in ''G'', there is an element ''a''′ in ''G'' with the property that ''a''′ * ''a'' = ''a'' * ''a''′ = ''e''. The element ''a''′ is an inverse of ''a'' with respect to *. ;Commutativity: For all ''a'', ''b'' in ''G'', ''a'' * ''b'' = ''b'' * ''a''. (詳細はTorsion (algebra)を参照) ;Torsion: A group ''G'' is a torsion group if every element in ''G'' is of finite order. ''G'' is torsion free if no element other than the identity is of finite order. 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「Torsion-free abelian group」の詳細全文を読む スポンサード リンク
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