Groups of homomorphisms of abelian groups

In summary, the conversation is about studying group homomorphisms and tensor products, specifically focusing on the groups ##Hom(\mathbb{Q}/\mathbb{Z},\mathbb{Q})## and ##Hom(\mathbb{Q},\mathbb{Q}/\mathbb{Z})##. The question at hand is to describe these groups of homomorphisms, with the additional information that ##\mathbb{Q}/\mathbb{Z}## and ##\mathbb{Q}## are both abelian groups and that ##\mathbb{Q}/\mathbb{Z}## is a finitely generated ##\mathbb{Z}##-module while ##\
  • #1
JD_Shadowplay
1
0
Hello everybody!
I've just started with studying group homorphisms and tensor products, so i am still not very sure if i undertstand the subject correct. I am stuck with a question and i would ask you for some help or hints how to proceed...

What i have to do is to describe ##Hom(\mathbb{Q}/\mathbb{Z},\mathbb{Q})## and ##Hom(\mathbb{Q},\mathbb{Q}/\mathbb{Z})##. I know that both ##\mathbb{Q}/\mathbb{Z}## and ##\mathbb{Q}## are abelian groups, ##\mathbb{Q}/\mathbb{Z}## as a ##\mathbb{Z}##-module is finitely generated, but ##\mathbb{Q}## as a ##\mathbb{Z}##-module is not finitely generated.
Can anybody help me with this problem? How is it meant "to describe the groups of homomorphisms"?
Thank you in advance!
 
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  • #2
A couple of things that may help
(Since Z is not a group under multiplication, you must be referring to addition)

- Q/Z is a torsion group. That is: every element is of finite order. Q has no elements of finite order.

- Try to find a set of Z generators of Q. Do the reciprocals of the prime numbers work?BTW: Why do you think Q/Z is a finitely generated Z module?
 
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1. What is a homomorphism of abelian groups?

A homomorphism of abelian groups is a function between two abelian groups that preserves the group operation. This means that for any two elements in the domain, their images under the function will have the same group operation as the original elements. In other words, the function maps between groups in a way that respects their group structure.

2. What is a group of homomorphisms of abelian groups?

A group of homomorphisms of abelian groups is a collection of homomorphisms between two abelian groups that form a group under function composition. This means that the group operation between two homomorphisms is defined as the composition of the two functions, and the identity element is the identity function. In other words, a group of homomorphisms is a set of functions that preserve the group structure and can be combined in a way that respects this structure.

3. How are groups of homomorphisms of abelian groups used in mathematics?

Groups of homomorphisms of abelian groups are used in various areas of mathematics, such as algebra, number theory, and topology. They are particularly useful in studying the structure and properties of abelian groups, as well as in understanding the relationships between different groups. They also have applications in fields such as cryptography, where groups of homomorphisms are used to encrypt and decrypt data.

4. What is the relationship between groups of homomorphisms of abelian groups and other algebraic structures?

Groups of homomorphisms of abelian groups have close relationships with other algebraic structures, such as rings and fields. In fact, a ring can be seen as a group of homomorphisms between two abelian groups, where the group operation is defined as addition and the composition of functions is defined as multiplication. Similarly, a field can be viewed as a group of homomorphisms between two abelian groups, where the group operation is defined as multiplication and the composition of functions is defined as division.

5. What are some important properties of groups of homomorphisms of abelian groups?

There are several important properties of groups of homomorphisms of abelian groups, such as associativity, commutativity, and the existence of an identity and inverse elements. These properties are inherited from the underlying abelian groups, as well as from the fact that the group operation is defined as function composition. Additionally, groups of homomorphisms can have finite or infinite order, and can also have subgroups and cosets just like other groups. Understanding these properties is crucial in studying and working with groups of homomorphisms of abelian groups.

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