|
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group ''A'' is the cardinality of a maximal linearly independent subset.〔Page 46 of 〕 The rank of ''A'' determines the size of the largest free abelian group contained in ''A''. If ''A'' is torsion-free then it embeds into a vector space over the rational numbers of dimension rank ''A''. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved. The term rank has a different meaning in the context of elementary abelian groups. == Definition == A subset of an abelian group is linearly independent (over Z) if the only linear combination of these elements that is equal to zero is trivial: if : where all but finitely many coefficients ''n''''α'' are zero (so that the sum is, in effect, finite), then all coefficients are 0. Any two maximal linearly independent sets in ''A'' have the same cardinality, which is called the rank of ''A''. Rank of an abelian group is analogous to the dimension of a vector space. The main difference with the case of vector space is a presence of torsion. An element of an abelian group ''A'' is classified as torsion if its order is finite. The set of all torsion elements is a subgroup, called the torsion subgroup and denoted ''T''(''A''). A group is called torsion-free if it has no non-trivial torsion elements. The factor-group ''A''/''T''(''A'') is the unique maximal torsion-free quotient of ''A'' and its rank coincides with the rank of ''A''. The notion of rank with analogous properties can be defined for modules over any integral domain, the case of abelian groups corresponding to modules over Z. 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「Rank of an abelian group」の詳細全文を読む スポンサード リンク
|