Free abelian group

In mathematics, a free abelian group is an abelian group, which has a base. This means that each element of the group can be written in exactly one way as a linear combination of elements of the base with integer coefficients.

In contrast to vector spaces, not every abelian group is a base, so there is a more specific concept of the free abelian group.

Note that a free abelian group is not the same as a free group which is abelian. In fact, most of the free groups are not abelian, and most of the free abelian groups are not free groups: a free abelian group is a free group if and even if their rank is at most. Therefore, some authors use to avoid misunderstandings, the term free abelian group in which the term is perceived free abelian as a single attribute.

Definition

The abelian group is called free over when every element of in exactly one way as a linear combination can represent over.

Here is a linear combination of a sum of the form of elements with integer coefficients. If the amount is infinite then one calls here in addition that only finitely many of the coefficients of zero may be different, so that the sum has a meaning.

Alternative definitions

The condition that the abelian group freely, can be divided into two parts:

• A generating system for the group, that is, each element of a linear combination on.

• Is free, that is the neutral element can be represented only on the trivial way as a linear combination over.

Every abelian group is a module in a natural way. Free abelian groups are therefore nothing more than free modules over.

Universal property

An abelian group is abelian if and only free with basis if it has the following universal property: If an arbitrary mapping of the set in an abelian group, then there exists a unique group homomorphism, which continues thus fulfilled for all.

This universal mapping property is to be equivalent to the above definition. Each of the two characterizations can therefore be used as a definition of free abelian groups. The respective other characterization is then an inference.

Examples

The set of integers is free abelian with basis.

The Cartesian product with componentwise addition is free abelian with basis.

General is free abelian with basis with the - th unit vector.

The set of sequences of integers, only finitely many of 0 different components, with the component-wise addition of a free abelian group; a base form the canonical unit vectors.

In contrast, the set of all sequences of integers with the component-wise addition, although an abelian group but not free abelian.

Finite abelian groups (except for the one-element group) are not free abelian groups.

Every free abelian group is torsion-free, but the reverse is not any torsion-free Abelian group also free abelian. For example, is not free abelian.

Construction

For every set can be a free abelian group with basis as follows can be constructed: We consider the set of all functions of the amount in the group of integers that take only a finite number of points of different values. This amount is an abelian group with the pointwise addition. We identify each element, with its characteristic function, ie with that function, which takes the place of the value and otherwise the value. Then is free abelian with basis.

The free abelian group on the set is unique in the following sense: If and two free abelian groups with basis, then they are canonically isomorphic, ie there exists an isomorphism with for all. This uniqueness makes it possible to speak of the free abelian group with basis.

Rank

Is an abelian group, both freely and clear about the amounts and then have the same cardinality. This is called rank of the free abelian group. According to the above construction, it is for each thickness up to isomorphism exactly one free abelian group of rank.

To prove that the rank is uniquely determined, one can proceed in several ways. For a free abelian group over a set of finite cardinality, this is accomplished particularly easily: With the universal mapping property of is the set of all group homomorphisms in the cyclic group of exactly elements. This is uniquely determined by the group.

In general, the rank of a free abelian group can be defined as the dimension of the vector space over a field (usually ). This dimension is uniquely determined by the group. This definition can also be used to all abelian groups ( whether free or not ) assign a rank, see Rank of an abelian group.

Base change and automorphisms

A free abelian group of rank has infinitely many bases. Each automorphism sends a base on a new basis. Conversely, there exists, for any two such bases and exactly one automorphism. Since every free abelian group is isomorphic to the rank, the automorphism group is isomorphic to the linear group. This already indicates, even if the free abelian groups are very easy to understand themselves, yet their automorphism groups are highly complicated and interesting.

Group homomorphisms and matrices

Free abelian groups have many nice features, similar to vector spaces and or generally free modules. For example, any group homomorphism can be represented as a finite rank matrix over between free abelian groups. For this purpose, a staple of and a basis of. The image in writes clearly than with coefficients. The number scheme and forms a matrix. Conversely, to each matrix in this way exactly is a group homomorphism. For addition and multiplication of matrices, the usual arithmetic rules apply, and these correspond to the addition and composition of homomorphisms. This leads to very efficient representations and methods of calculation.

Subgroups

In a free abelian group every subgroup is free abelian. This is not self-evident and does not apply generally for modules over rings. ( About the polynomial ring, for example, is a free module with basis, but the sub-module is not free. )

Moreover, the rank of a subgroup of a free abelian group is always less than or equal to the rank of the entire group. This is not, of course, not true for free groups. ( For instance containing the free group of rank subsets of each rank. )

The subgroups of a free abelian group of rank can be classified as follows. Each subgroup has rank, and there is a basis of and integers so that a base of being. This can be proved by means of the Gaussian algorithm for integer matrices.

Application on finitely generated abelian groups

Free Abelian groups play an important role in the classification of finitely generated abelian groups. Every finitely generated abelian group is the homomorphic image of a free abelian group, ie a epimorphism. The core is a free abelian group again and there is a basis of and integers so that a base of being. From this display, you see immediately a group isomorphism.