# Why are the integers and rationals not isomorphic under addition?

In summary, the integers are not isomorphic to the rationals. This is due to the fact that the rationals cannot be generated, and because there is an a such that <a> = Q.

## Homework Statement

Prove that the integers (under addition) are not isomorphic to the rationals (under addition).

## Homework Equations

Two groups are isomorphic if there is an isomorphism between them.

If there is an isomorphism from G to H, f : G --> H, then G is cyclic iff H is cyclic.

A group G is cyclic if $$\{ x^n | n \in \mathbb{Z} \} = G, for some x \in G$$ .

## The Attempt at a Solution

The integers are generated by $$<1>$$. We can show that Z and Q are not isomorphic if we show that the rationals cannot be generated. Thus assume they are. Then there is an a such that

[tex] <a> = Q [\tex].
[tex] 0a = 0, 1a = \frac{l}{m} , 2a = \fract{2l}{m}[\tex].

Because the rationals are dense there is a [tex] b \in Q s.t. \frac{l}{m} < b < \frac{2l}{m} [\tex]

We must show that [tex] b = ka = \frac{kl}{m}, thus \frac{l}{m} < \frac{kl}{m} < \frac{2l}{m} [\tex].

Now I don't know what to do. The above is not a contradiction. Any ideas?

Last edited:
Suppose Q is cyclic.

Let p/q be a generator.

Can you find a rational number which is not an integer multiple of p/q?
Think of a rational involving p and q somehow.

samkolb said:
Suppose Q is cyclic.

Let p/q be a generator.

Can you find a rational number which is not an integer multiple of p/q?
Think of a rational involving p and q somehow.

sure, continuing

Observe p/2q

Then,

p/2q = p/q where p,q are not equal to zero. (if p is zero then the set is finite).

From the above conclude that
pq=2pq, therefore 1=2.
done.

No, no. <p/q> is the set of number k*p/q where k is an integer. Set kp/q=p/(2q) and derive a different contradiction.

Dick said:
No, no. <p/q> is the set of number k*p/q where k is an integer. Set kp/q=p/(2q) and derive a different contradiction.

Thanks! Yeah, I don't know what I was thinking there. I went and got some food, came back and hit myself in the head on that one.

## 1. What does it mean for two structures to be non-isomorphic?

Non-isomorphism refers to two structures that do not have the same underlying organization or arrangement of elements. This means that while the two structures may share similar properties or characteristics, they are not identical and cannot be transformed into one another through a series of operations.

## 2. How can you prove that Z and Q are not isomorphic?

To prove that Z (the set of integers) and Q (the set of rational numbers) are not isomorphic, one can use the definition of isomorphism which states that two structures are isomorphic if there exists a one-to-one mapping between them that preserves the structure. Since Z and Q have different properties and operations, it is not possible to find a mapping that satisfies this condition and therefore they are not isomorphic.

## 3. What are some key differences between Z and Q that make them non-isomorphic?

One key difference between Z and Q is the presence of fractions and decimal numbers in Q that are not present in Z. Additionally, Q has the property of closure under multiplication and division, while Z does not. These differences in properties and operations prevent the two structures from being isomorphic.

## 4. Can two structures be non-isomorphic but still have similar elements?

Yes, two structures can have similar elements or properties but still be non-isomorphic. This is because isomorphism is determined by the underlying organization and arrangement of elements, not just the individual elements themselves.

## 5. Are there any real-world applications of understanding non-isomorphic structures?

Yes, understanding non-isomorphic structures is important in various fields such as computer science, mathematics, and chemistry. In computer science, it is used in cryptography to create secure encryption methods. In mathematics, it is used to classify different types of structures. In chemistry, it is used to determine the isomerism of molecules, which has important implications for their chemical properties and reactions.

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