Free Abelian group

In mathematics , a free Abelian group is an Abelian group that has a base as a module . ${\displaystyle \mathbb {Z} }$

In contrast to vector spaces , not every Abelian group has a basis, which is why there is the more specific term of the free Abelian group.

Note that a free Abelian group is not the same as a free group that is Abelian . In fact, most free groups are non- Abelian, and most free Abelian groups are not free groups: a free Abelian group is a free group if and only if its rank is at most . To avoid misunderstandings, some authors use the term free Abelian group, in which the term free Abelian is understood as a single attribute. ${\displaystyle 1}$

definition

The Abelian group is about freely when a base of the - module is. This means that each element of can be represented in exactly one way as a linear combination via . ${\displaystyle F}$ ${\displaystyle B\subset F}$ ${\displaystyle B}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle B}$

Here is a -linear combination over a sum of the shape of elements from with integer coefficients . If the set is infinite, one also demands that only a finite number of the coefficients may differ from zero so that the sum has a meaning. ${\displaystyle \mathbb {Z} }$ ${\displaystyle B}$ ${\displaystyle \textstyle \sum _{b\in B}\lambda _{b}\cdot b}$ ${\displaystyle B}$ ${\displaystyle \lambda _{b}\in \mathbb {Z} }$ ${\displaystyle B}$ ${\displaystyle \lambda _{b}}$

The elements of the free Abelian group generated by are also referred to as formal sums of elements from . For example, the formal sums of singular simplices are used in the definition of singular homology or the formal sums of complex numbers are used in the definition of the Bloch group . ${\displaystyle B}$ ${\displaystyle B}$

Alternative definitions

The condition that the Abelian group is free can be divided into two parts: ${\displaystyle F}$ ${\displaystyle B}$

• ${\displaystyle B}$is a generating system for the group , that is, every element of is a linear combination over . ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle B}$

• ${\displaystyle B}$is free, that is, the neutral element can only be represented in the trivial way as -linear combination via . ${\displaystyle 0}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle B}$

Every Abelian group is naturally a module. Free Abelian groups are therefore nothing more than free modules over . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$

Universal property

An Abelian group is free Abelian with a basis if and only if it has the following universal property : If an arbitrary mapping of the set into an Abelian group , then there is exactly one group homomorphism which continues, i.e. fulfills for all . ${\displaystyle F}$ ${\displaystyle B\subset F}$ ${\displaystyle f\colon B\to A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle h\colon F\to A}$ ${\displaystyle f}$ ${\displaystyle h(b)=f(b)}$ ${\displaystyle b\in B}$

This universal mapping property is equivalent to the above definition. Either of the two characterizations can therefore be used as a definition of free Abelian groups. The other characterization is then a consequence.

Examples

The group of integers is freely Abelian with a base . ${\displaystyle (\mathbb {Z} ,+)}$ ${\displaystyle \{1\}}$

The Cartesian product with component addition is free Abelian with a base . ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$ ${\displaystyle \{(1,0),(0,1)\}}$

Generally, free abelian with base wherein the -th unit vector. ${\displaystyle \mathbb {Z} ^{r}}$ ${\displaystyle \{e_{1},\dotsc ,e_{r}\}}$ ${\displaystyle e_{i}=(0,\dotsc ,0,1,0,\dotsc ,0)}$ ${\displaystyle i}$

The set of sequences of integers that only have finitely many components different from 0 is, with the component-wise addition, a free Abelian group; the canonical unit vectors form a basis . ${\displaystyle \mathbb {Z} ^{(\mathbb {N} )}}$ ${\displaystyle (0,\dotsc ,0,1,0,\dotsc )}$

On the other hand, the set of all sequences of whole numbers with the component-wise addition is an Abelian group, but not free Abelian. ${\displaystyle \mathbb {Z} ^{\mathbb {N} }}$

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

Every free Abelian group is torsion-free , but conversely, not every torsion-free Abelian group is also free Abelian. For example, Abelian is not free. ${\displaystyle (\mathbb {Q} ,+)}$

construction

For every set a free Abelian group with a basis can be constructed as follows: We consider the set of all functions of the set in the group of integers , which only take on different values in finitely many places . This set is an Abelian group with pointwise addition. We identify each item with its characteristic function, so with that function , which at the point the value of accepting and otherwise the value . Then free is abelian with a base . ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle F(B)=\mathbb {Z} ^{(B)}}$ ${\displaystyle B\to \mathbb {Z} }$ ${\displaystyle B}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle 0}$ ${\displaystyle b\in B}$ ${\displaystyle B\to \mathbb {Z} }$ ${\displaystyle b}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle F(B)}$ ${\displaystyle B}$

The free Abelian group over the set is unique in the following sense: If and are two free Abelian groups with a base , then they are canonically isomorphic , that is, there is exactly one isomorphism with for all . This unambiguity allows one to speak of the free Abelian group with a basis . ${\displaystyle B}$ ${\displaystyle F_{1}}$ ${\displaystyle F_{2}}$ ${\displaystyle B}$ ${\displaystyle h\colon F_{1}\to F_{2}}$ ${\displaystyle h(b)=b}$ ${\displaystyle b\in B}$ ${\displaystyle B}$

rank

If an Abelian group is both free above and free above , then the sets and have the same cardinality . This is called the rank of the free Abelian group . According to the above construction, there is exactly one free Abelian group of rank for every thickness apart from isomorphism . ${\displaystyle F}$ ${\displaystyle B}$ ${\displaystyle B'}$ ${\displaystyle B}$ ${\displaystyle B'}$ ${\displaystyle F}$ ${\displaystyle n}$ ${\displaystyle n}$

There are several ways of proving that rank is uniquely determined. This is particularly easy for a free Abelian group over a set of finite thickness : Due to the universal mapping property of , the set of all group homomorphisms in the cyclic group consists of exactly elements. This is clearly defined by the group . ${\displaystyle F=F(B)}$ ${\displaystyle B}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle F}$ ${\displaystyle Hom(F,C_{2})}$ ${\displaystyle C_{2}}$ ${\displaystyle 2^{n}}$ ${\displaystyle n}$ ${\displaystyle F}$

In general, the rank of a free Abelian group can be defined as the dimension of the vector space over a body (usually ). This dimension is clearly determined by the group . This definition can also be used to assign a rank to all Abelian groups (whether free or not), see Rank of an Abelian Group . ${\displaystyle F}$ ${\displaystyle F\otimes K}$ ${\displaystyle K}$ ${\displaystyle K=\mathbb {Q} }$ ${\displaystyle F}$

Change of base and automorphisms

A free Abelian group of rank has infinitely many bases. Each automorphism sends one base to a new base . Conversely, there is two such bases and exactly one automorphism . Since every free Abelian group is of rank to isomorphic, the automorphism group is isomorphic to the linear group . This already indicates: Even if the free Abelian groups themselves are very easy to understand, their automorphism groups are highly complex and interesting. ${\displaystyle F}$ ${\displaystyle r\geq 2}$ ${\displaystyle h\colon F\to F}$ ${\displaystyle B=(b_{1},\dotsc ,b_{r})}$ ${\displaystyle B'=(h(b_{1}),\dotsc ,h(b_{r}))}$ ${\displaystyle B}$ ${\displaystyle B'}$ ${\displaystyle h\colon F\to F}$ ${\displaystyle F}$ ${\displaystyle r}$ ${\displaystyle \mathbb {Z} ^{r}}$ ${\displaystyle \mathrm {Aut} (F)}$ ${\displaystyle \mathrm {GL} _{r}(\mathbb {Z} )}$

Group homomorphisms and matrices

Free Abelian groups have many convenient properties, similar to vector spaces and / or free modules in general. For example, every group homomorphism between free Abelian groups of finite rank can be represented as a matrix over . For this, let a basis of and a basis of . The picture in is clearly written as having coefficients . The number scheme with and forms a matrix. Conversely, each matrix corresponds exactly to one group homomorphism in this way. The usual calculation rules apply to the addition and multiplication of matrices, and these correspond to the addition and composition of homomorphisms. This leads to very efficient representations and calculation methods. ${\displaystyle h\colon F\to G}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle (f_{1},\dotsc ,f_{s})}$ ${\displaystyle F}$ ${\displaystyle (g_{1},\dotsc ,g_{r})}$ ${\displaystyle G}$ ${\displaystyle h(f_{j})}$ ${\displaystyle G}$ ${\displaystyle h(f_{j})=a_{1j}g_{1}+\dotsb +a_{rj}g_{r}}$ ${\displaystyle a_{ij}\in \mathbb {Z} }$ ${\displaystyle (a_{ij})}$ ${\displaystyle i=1,\dotsc ,r}$ ${\displaystyle j=1,\dotsc ,s}$ ${\displaystyle r\times s}$

Subgroups

In a free Abelian group , each subgroup is free Abelian . This is by no means self-evident and does not apply generally to modules over rings. (For example, above the polynomial ring there is a free module with a base , but the sub-module is not free.) ${\displaystyle F}$ ${\displaystyle U\subset F}$ ${\displaystyle \mathbb {Z} [X]}$ ${\displaystyle \mathbb {Z} [X]}$ ${\displaystyle 1}$ ${\displaystyle (2,X)}$

In addition, the rank of a subgroup of a free Abelian group is always less than or equal to the rank of the entire group . This cannot be taken for granted and does not apply to free groups . (For example, the free group of rank contains subsets of each rank .) ${\displaystyle U\subset F}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle 2}$ ${\displaystyle r\in \mathbb {N} }$

The subgroups of a free Abelian group of rank can be classified as follows. Each subgroup has rank with , and there is a base of and integers such that there is a base of . This can be proven using the Gaussian algorithm for integer matrices. ${\displaystyle F}$ ${\displaystyle r}$ ${\displaystyle U\subset F}$ ${\displaystyle s}$ ${\displaystyle 0\leq s\leq r}$ ${\displaystyle (b_{1},\dotsc ,b_{r})}$ ${\displaystyle F}$ ${\displaystyle k_{1},\dotsc ,k_{s}\in \mathbb {Z} \setminus \{0\},}$ ${\displaystyle (k_{1}b_{1},\dotsc ,k_{s}b_{s})}$ ${\displaystyle U}$

Application to 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, i.e. an epimorphism . The core is again a free Abelian group and there is a base of and integers such that there is a base of . A group isomorphism is obtained directly from this representation . ${\displaystyle A}$ ${\displaystyle h\colon \mathbb {Z} ^{r}\to A}$ ${\displaystyle (b_{1},\dotsc ,b_{r})}$ ${\displaystyle \mathbb {Z} ^{r}}$ ${\displaystyle k_{1},\dotsc ,k_{s}\in \mathbb {Z} \setminus \{0\},}$ ${\displaystyle (k_{1}b_{1},\dotsc ,k_{s}b_{s})}$ ${\displaystyle \ker(h)}$ ${\displaystyle A\cong \mathbb {Z} /k_{1}\oplus \dotsb \oplus \mathbb {Z} /k_{s}\oplus \mathbb {Z} ^{r-s}}$

literature

• Gisbert Wüstholz : Algebra. For students of mathematics, physics, computer science. Vieweg, Wiesbaden 2004, ISBN 3-528-07291-1 .