Direct sum of free abelian groups

In summary, the direct sum of a family of free abelian groups is a free abelian group, and this is only true when the groups are finitely generated. The disjoint union of the basis of the groups is a basis for the direct sum.
  • #1
mathgirl1
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Show the direct sum of a family of free abelian groups is a free abelian group.

My first thought was to just say that since each group is free abelian we know it has a non empty basis. Then we can take the direct sum of the basis to be the basis of the direct sum of a family of free abelian groups. But not sure it makes sense to say direct sum of basis.

Is it just as simple that the direct sum of a family of free abelian groups is isomorphic to the direct sum of the additive group of integers?

I am probably making this way harder than it is but I am not sure how to state the obvious either. Any help is appreciated. Thanks!
 
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  • #2
Hi mathgirl,

mathgirl said:
Is it just as simple that the direct sum of a family of free abelian groups is isomorphic to the direct sum of the additive group of integers?

No, this is only true when your groups are finitely generated.

Try to prove that the disjoint union of the basis of your groups is a basis for the direct sum. (Probably you meant some like that when talking abaout "direct sum of basis")
 

FAQ: Direct sum of free abelian groups

What is the direct sum of free abelian groups?

The direct sum of free abelian groups is a mathematical operation that combines two or more free abelian groups to form a new group. It is denoted by a plus sign (+) between the groups, and the resulting group is called the direct sum of the original groups.

How is the direct sum of free abelian groups different from the direct product?

The direct sum and direct product of free abelian groups are similar in that they both combine multiple groups to form a new group. However, the direct sum only includes elements that are non-zero in each individual group, while the direct product includes all possible combinations of elements from each group.

What are some properties of the direct sum of free abelian groups?

Some properties of the direct sum of free abelian groups include: commutativity, associativity, distributivity, and the existence of an identity element (the zero element). Additionally, the direct sum is isomorphic to the direct product, meaning they have the same structure and behave in the same way.

How is the direct sum of free abelian groups used in mathematics?

The direct sum of free abelian groups is used in various areas of mathematics, such as algebra, number theory, and topology. It is particularly useful in abstract algebra for studying the structure and properties of groups, and in number theory for understanding the structure of abelian groups.

Are there any applications of the direct sum of free abelian groups in real-world problems?

Yes, the direct sum of free abelian groups has applications in real-world problems, particularly in the field of coding theory. It is used to create error-correcting codes, which are used in data transmission and storage to ensure the accuracy of transmitted information.

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