Infinitely generated and isomorphism

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In summary, the conversation discusses the groups Q+ and Q*, and how to prove that neither of these groups is finitely generated using the fact that there are infinitely many primes. It also explores the idea of proving that Q+ is not isomorphic to Q* by considering their subgroup structure.
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doongly
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Homework Statement


Consider the groups Q+ and Q* (rational under addition and ration under multiplication). Prove that neither of these groups is finitely geneated by using the fact that there are infinitely many primes. And prove that Q+ is not isomorphic to Q*.


2. The attempt at a solution
I know that for Q+ if we pick coprimes to be the denominators; then 1/w cannot be generated by rationals. Is this a proof? But I have no idea about Q* and the isomorphism.
 
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  • #2
What you said for [tex]\mathbb{Q}^+[/tex] is too sketchy for me to guess whether you have the right idea or not.

For [tex]\mathbb{Q}^\times[/tex], think about how to generate the primes.

For the isomorphism: one way to see that two groups are not isomorphic is to show that they have different subgroup structure.
 

Related to Infinitely generated and isomorphism

1. What does it mean for a group to be infinitely generated?

Infinitely generated refers to a group that has an infinite number of generators, meaning that it can be generated by an infinite number of elements. This means that there is no finite list of elements that can generate the entire group.

2. How does a group being infinitely generated affect its structure?

Being infinitely generated typically means that a group has more complex and diverse structure compared to a finitely generated group. This can make it more difficult to understand and study, as there are infinitely many possibilities for how the group can behave.

3. What is an isomorphism between infinitely generated groups?

An isomorphism is a bijective function between two groups that preserves their group structure. In the case of infinitely generated groups, an isomorphism must also preserve the infinite number of generators and their relations.

4. Can two infinitely generated groups be isomorphic?

Yes, it is possible for two infinitely generated groups to be isomorphic. This means that they have the same group structure and can be mapped onto each other using an isomorphism.

5. How are infinitely generated groups used in mathematics and science?

Infinitely generated groups have many applications in mathematics and science, particularly in abstract algebra, topology, and group theory. They are also used in modeling and analyzing complex systems, such as in physics and computer science.

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