Section 67: Problem 1 Solution »

Section 67: Direct Sums of Abelian Groups

is an abelian group.
Subgroups , , generate a group if every element of can be expressed as a finite sum of elements of the groups : . If for each element of the representation where for all but finite number of indices is unique, is called the direct sum of the groups : .
  • The sum is direct iff has a unique solution.
(extension condition for internal direct sums) If is an abelian group, and is a family of its subgroups that generate , then
is the direct sum of
iff
given any abelian group , and any family of homomorphisms , there is a homomorphism such that for every , .
Moreover, such is unique.
  • The direct sum of direct sums is a direct sum, in particular, is associative.
  • If , then is isomorphic to .
Given a family of abelian groups , an abelian group is called the external direct sum of (relative to monomorphisms ) if there are monomorphisms such that .
  • There is always such a group , 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 is an abelian group, is a family of abelian groups, and is a family of homomorphisms such that generate , then
each is a monomorphism and is the direct sum of
iff
given any abelian group , and any family of homomorphisms , there is a homomorphism such that for every , .
Moreover, such is unique.
Elements of an abelian group are said to generate the group if they generate subgroups that generate . If each is infinite cyclic, and is the direct sum of , then is said to be a free abelian group having as a basis. If the number of elements in a basis of is finite, then this number is called the rank of .
  • The rank of is unique.
  • Every subgroup of a free abelian group of rank is a free abelian group of rank at most .
  • The torsion subgroup of is the set of all elements of having a finite order ( for some ).\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 is an abelian group, and is a family of its elements that generate , then
is a free abelian group with basis
iff
given any abelian group , and any family of elements of , there is a homomorphism such that for every , .
Moreover, such is unique.