In
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 com ...
''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
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...
subset. The rank of ''A'' determines the size of the largest
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a su ...
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 ...
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 ''A'' is
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...
(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 coor ...
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 ...
. 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 d ...
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
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, 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 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 sums:
::
: 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
Indecomposability or indecomposable may refer to any of several subjects in mathematics:
* Indecomposable module, in algebra
* Indecomposable distribution, in probability
* Indecomposable continuum, in topology
* Indecomposability (intuitionist ...
, 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
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
''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 (mathematics), 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 e ...
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 ...
) 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