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Concept
Rank of an abelian group
Summary
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. 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.
A subset {aα} of an abelian group A 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 nα are zero (so that the sum is, in effect, finite), then all coefficients are zero. Any two maximal linearly independent sets in A have the same cardinality, which is called the rank of A.
The 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. For this, see finitely generated module#Generic rank.
The rank of an abelian group A coincides with the dimension of the Q-vector space A ⊗ Q. If A is torsion-free then the canonical map A → A ⊗ Q is injective and the rank of A is the minimum dimension of Q-vector space containing A as an abelian subgroup.
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