In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the rank, Prüfer rank, or torsion-free rank of an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
''A'' is the
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over the
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
of dimension rank ''A''. For
finitely generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, ...
s, rank is a strong invariant and every such group is determined up to isomorphism by its rank and
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
.
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 group
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian grou ...
s.
Definition
A subset 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 ''n''
''α'' 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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
, which is called the rank of ''A''.
The rank of an abelian group is analogous to the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. 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
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
is finite. The set of all torsion elements is a subgroup, called the
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
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
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over any
integral domain, the case of abelian groups corresponding to modules over Z. For this, see
finitely generated module#Generic rank.
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
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
and the rank of ''A'' is the minimum dimension of Q-vector space containing ''A'' as an abelian subgroup. In particular, any intermediate group Z
''n'' < ''A'' < Q
''n'' 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.
[. O]
p. 46
Thomas and Schneider refer to "...this failure to classify even the rank 2 groups in a satisfactory way..."
* Rank is additive over
short exact sequences: if
::
: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 sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
s:
::
: 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
, there is a torsion-free abelian group of rank
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
, there exists a torsion-free abelian group ''A'' of rank ''n'' such that for any partition
into ''k'' natural summands, the group ''A'' is the direct sum of ''k'' indecomposable subgroups of ranks
. 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 ''A''
''n'' is isomorphic to ''B''
''n'' 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 ''R''
0, the
quotient field, of the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of the module with the field:
::
It makes sense, since ''R''
0 is a field, and thus any module (or, to be more specific,
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
) over it is free.
It is a generalization, since every 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'',
::
See also
*
Rank of a group
References
{{reflist
Abelian group theory