Q (the rationals) not free abelian yet iso to Z^2?

Thread starter TopCat
Start date Nov 25, 2011

In summary, Q (the rationals) not being free abelian means that it cannot be generated by a single element, unlike some other groups such as Z (the integers). However, there exists a bijective homomorphism between Q and Z^2, showing that they have the same underlying structure. Isomorphism between two groups means they are equivalent and can be thought of as the same group with different elements. The lack of a basis in Q affects its isomorphism to Z^2 by requiring multiple elements to generate the elements in Z^2. Understanding this isomorphism has applications in fields such as algebraic geometry and topology, providing insights into the structure of different groups.

Nov 25, 2011

TopCat

So I just proved that Q (the rationals) is not free abelian since it must have an empty basis. But Q is isomorphic to Z^2 = ƩZ which is equivalent to Q having a nonempty basis. I must be wrong about the isomorphism but I don't see why.

Last edited: Nov 25, 2011

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Nov 25, 2011

mathwonk

Science Advisor

Homework Helper

there is a map ZxZ*-->Q but it isn't injective.

Nov 25, 2011

TopCat

D'oh! Thanks, mathwonk.

1. What does it mean for Q (the rationals) to not be free abelian?

Being free abelian means that a group has a basis, or a set of elements that can be used to generate all other elements in the group. In the case of Q, this means that it cannot be generated by a single element, unlike some other groups such as Z (the integers).

2. How is Q isomorphic to Z^2 (the direct product of two copies of Z)?

There exists a bijective homomorphism, or a function that preserves the group structure, between Q and Z^2. This means that although the two groups may have different elements, they have the same underlying structure and can be viewed as essentially the same group.

3. Can you explain the concept of isomorphism in more detail?

An isomorphism between two groups means that they have the same structure, even if the individual elements may be different. It essentially shows that the two groups are equivalent and can be thought of as the same group with different elements.

4. How does the lack of a basis in Q affect its isomorphism to Z^2?

Since Q does not have a basis, it cannot be generated by a single element. This means that the bijective homomorphism between Q and Z^2 must take into account multiple elements in Q to generate the elements in Z^2. This is why Q is not free abelian, but still isomorphic to Z^2.

5. What are some real-world applications of understanding the isomorphism between Q and Z^2?

Understanding the isomorphism between Q and Z^2 can be useful in various areas of mathematics and physics, such as in the study of algebraic geometry and topology. It can also help in understanding the underlying structure of different groups and how they can be related to each other, leading to new insights and discoveries in these fields.