In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to "contain" it; or alternatively how large a free abelian group it can contain as a subgroup. Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ...

Definition

An abelian group is often thought of as composed of its torsion subgroup T, and its torsion-free part A/T. The t.f. rank describes how complicated the torsion-free part can be. In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ...

More precisely, let A be an abelian group and T the torsion subgroup, T = { a in A : na = 0 for some nonzero integer n }. Let Q denote the set of rational numbers. The t.f. rank of A is equal to all of the following cardinal numbers: In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...

• The vector space dimension of the tensor product of the abelian groups Q and A
• The vector space dimension of the smallest Q-vector space containing the torsion-free group A/T
• The largest cardinal d such that A contains a copy of the direct sum of d copies of the integers Z
• The cardinality of a maximal Z-linearly independent subset of A

Following the same pattern, we may also define t.f. ranks of all modules over any principal ideal domain R. Instead of Q we then use the field of fractions of R. In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In abstract algebra, a module is a generalization of a vector space. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ...

Properties

There are many abelian groups of rank 0, but the only torsion-free one is the trivial group {0}. In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ...

As one would expect, the rank of Zⁿ is n for every natural number n. More generally, the rank of any free abelian group (as explained in that article) coincides with its t.f. rank. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ...

The following fact can often be used to compute ranks: if

is a short exact sequence of abelian groups, then In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

(Proof: tensoring the given sequence with Q yields a short exact sequence of Q-vector spaces since Q is flat; vector space dimensions are additive on short exact sequences.) In abstract algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. ...

Another useful formula, familiar from vector space dimensions, is the following about arbitrary direct sums: In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

Curiosities about large rank groups

There is a complete classification of t.f. rank 1 torsion-free groups. Larger ranks are more difficult to classify, and no current system of classifying rank 2 torsion-free groups is considered very effective.

Larger ranks, especially infinite ranks, are often the source of entertaining paradoxical groups. For instance for every cardinal d, there are many torsion-free abelian groups of rank d that cannot be written as a direct sum of any pair of their proper subgroups. Such groups are called indecomposable, since they are not simply built up from other smaller groups. These examples show that torsion-free rank 1 groups (which are relatively well understood) are not the building blocks of all abelian groups.

Furthermore, for every integer n ≥ 3, there is a rank 2n-2 torsion-free abelian group that is simultaneously a sum of two indecomposable groups, and a sum of n indecomposable groups. Hence for ranks 4 and up, even the number of building blocks is not well-defined.

Another example, due to A.L.S. Corner, shows that the situation is as bad as one could possibly imagine: Given integers n ≥ k ≥ 1, there is a torsion-free group A of rank n, such that for any partition of n into r₁ + ... + r_k = n, each r_i being a positive integer, A is the direct sum of k indecomposable groups, the first with rank r₁, the second r₂, ..., the k-th with rank r_k. This shows that a single group can have all possible combinations of a given number of building blocks, so that even if one were to know complete decompositions of two torsion-free groups, one would not be sure that they were not isomorphic. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

Other silly 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.

When one allows infinite rank, one is treated to a group G contained in a group K such that K is indecomposable and is generated by G and a single element, and yet every nonzero direct summand of G has yet another nonzero direct summand.