Rank of an abelian group

In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.^[1] 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 {a_α} 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

\sum_\alpha n_\alpha a_\alpha = 0, \quad n_\alpha\in\mathbb{Z},

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.

Properties

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. In particular, any intermediate group Zⁿ < A < Qⁿ has rank n.

Abelian groups of rank 0 are exactly the periodic abelian groups.

The group Q of rational numbers has rank 1. Torsion-free abelian groups of rank 1 are realized as subgroups of Q and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2.^[2]

Rank is additive over short exact sequences: if

0\to A\to B\to C\to 0\;

is a short exact sequence of abelian groups then rk B = rk A + rk C. This follows from the flatness of Q and the corresponding fact for vector spaces.

Rank is additive over arbitrary direct sums:

\operatorname{rank}\left(\bigoplus_{j\in J}A_j\right) = \sum_{j\in J}\operatorname{rank}(A_j),

where the sum in the right hand side uses cardinal arithmetic.

Groups of higher rank

Abelian groups of rank greater than 1 are sources of interesting examples. For instance, for every cardinal d there exist torsion-free abelian groups of rank d that are indecomposable, i.e. cannot be expressed as a direct sum of a pair of their proper subgroups. These examples demonstrate that torsion-free abelian group of rank greater than 1 cannot be simply built by direct sums from torsion-free abelian groups of rank 1, whose theory is well-understood. Moreover, for every integer n ≥ 3, there is a torsion-free abelian group of rank 2n − 2 that is simultaneously a sum of two indecomposable groups, and a sum of n indecomposable groups. Hence even the number of indecomposable summands of a group of an even rank greater or equal than 4 is not well-defined.

Another result about non-uniqueness of direct sum decompositions is due to A.L.S. Corner: given integers n ≥ k ≥ 1, there exists a torsion-free abelian group A of rank n such that for any partition n = r₁ + ... + r_k into k natural summands, the group A is the direct sum of k indecomposable subgroups of ranks r₁, r₂, ..., r_k. Thus the sequence of ranks of indecomposable summands in a certain direct sum decomposition of a torsion-free abelian group of finite rank is very far from being an invariant of A.

Other surprising examples include torsion-free rank 2 groups A_n,m and B_n,m such that Aⁿ is isomorphic to Bⁿ if and only if n is divisible by m.

For abelian groups of infinite rank, there is an example of a group K and a subgroup G such that

K is indecomposable;
K is generated by G and a single other element; and
Every nonzero direct summand of G is decomposable.

Generalization

The notion of rank can be generalized for any module M over an integral domain R, as the dimension over R₀, the quotient field, of the tensor product of the module with the field:

\text{rank} (M)=\dim_{R_0} M\otimes_R R_0

It makes sense, since R₀ is a field, and thus any module (or, to be more specific, vector space) over it is free.

It is a generalization, since any abelian group is a module over the integers. It easily follows that the dimension of the product over Q is the cardinality of maximal linearly independent subset, since for any torsion element x and any rational q

x\otimes_{\mathbf Z} q = 0

References

↑ Page 46 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001
↑ Thomas, Simon; Schneider, Scott (2012), "Countable Borel equivalence relations", in Cummings, James; Schimmerling, Ernest, Appalachian Set Theory: 2006-2012, London Mathematical Society Lecture Note Series 406, Cambridge University Press, pp. 25–62, doi:10.1017/CBO9781139208574.003, ISBN 9781107608504 . On p. 46, Thomas and Schneider refer to "...this failure to classify even the rank 2 groups in a satisfactory way..."

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