Section 67: Direct Sums of Abelian Groups

$(G,+)$ is an abelian group.

Subgroups $G_{\alpha}$ , $\alpha\in J$ , generate a group $G$ if every element of $G$ can be expressed as a finite sum of elements of the groups $G_{\alpha}$ : $G=\sum_{\alpha\in J}G_{\alpha}$ . If for each element $x$ of $G$ the representation $x=\sum_{\alpha\in J}x_{\alpha}$ where $x_{\alpha}=0$ for all but finite number of indices is unique, $G$ is called the direct sum of the groups $G_{\alpha}$ : $G=\bigoplus_{\alpha\in J}G_{\alpha}$ .

The sum $G=\sum_{\alpha\in J}G_{\alpha}$ is direct iff $\sum_{\alpha\in J}x_{\alpha}=0$ has a unique solution.

(extension condition for internal direct sums) If $G$ is an abelian group, and $\{G_{\alpha}\}_{\alpha\in J}$ is a family of its subgroups that generate $G$ , then

$G$ is the direct sum of $G_{\alpha}$

iff

given any abelian group $H$ , and any family of homomorphisms $\{h_{\alpha}:G_{\alpha}\to H\}_{\alpha\in J}$ , there is a homomorphism $h:G\to H$ such that for every $\alpha\in J$ , $h|_{G_{\alpha}}=h_{\alpha}$ .

Moreover, such $h$ is unique.

The direct sum of direct sums is a direct sum, in particular, $\bigoplus$ is associative.
If $G=G_{1}\bigoplus G_{2}$ , then $G/G_{2}$ is isomorphic to $G_{1}$ .

Given a family of abelian groups $\{G_{\alpha}\}_{\alpha\in J}$ , an abelian group $G$ is called the external direct sum of $G_{\alpha}$ (relative to monomorphisms $i_{\alpha}$ ) if there are monomorphisms $i_{\alpha}:G_{\alpha}\to G$ such that $G=\bigoplus_{\alpha\in J}i_{\alpha}(G_{\alpha})$ .

There is always such a group $G$ , hence, there is always the external direct sum of any family of abelian groups.
The external direct sum is defined uniquely up to isomorphism.

(extension condition for external direct sums) If $G$ is an abelian group, $\{G_{\alpha}\}_{\alpha\in J}$ is a family of abelian groups, and $\{i_{\alpha}:G_{\alpha}\to G\}$ is a family of homomorphisms such that $i_{\alpha}(G_{\alpha})$ generate $G$ , then

each $i_{\alpha}$ is a monomorphism and $G$ is the direct sum of $i_{\alpha}(G_{\alpha})$

iff

Moreover, such $h$ is unique.

Elements $\{a_{\alpha}\}_{\alpha\in J}$ of an abelian group $G$ are said to generate the group $G$ if they generate subgroups $G_{\alpha}$ that generate $G$ . If each $G_{\alpha}$ is infinite cyclic, and $G$ is the direct sum of $G_{\alpha}$ , then $G$ is said to be a free abelian group having $\{a_{\alpha}\}_{\alpha\in J}$ as a basis. If the number of elements in a basis of $G$ is finite, then this number is called the rank of $G$ .

The rank of $G$ is unique.
Every subgroup of a free abelian group of rank $n$ is a free abelian group of rank at most $n$ .
The torsion subgroup of $G$ is the set of all elements $a$ of $G$ having a finite order ( $ma=0$ for some $m>0$ ).\begin_inset Separator latexpar\end_inset
- The torsion subgroup of a free abelian group is trivial.
- There are abelian groups that have trivial torsion subgroups, but which are not free.

(extension condition for free abelian groups) If $G$ is an abelian group, and $\{a_{\alpha}\}_{\alpha\in J}$ is a family of its elements that generate $G$ , then

$G$ is a free abelian group with basis $\{a_{\alpha}\}_{\alpha\in J}$

iff

given any abelian group $H$ , and any family $\{y_{\alpha}\}_{\alpha\in J}$ of elements of $H$ , there is a homomorphism $h:G\to H$ such that for every $\alpha\in J$ , $h(a_{\alpha})=y_{\alpha}$ .

Moreover, such $h$ is unique.

Subpages

Section 67: Direct Sums of Abelian Groups

Section 67: Direct Sums of Abelian Groups