1. ## Monoids and Groups

Can I get help on the following problem:
Is the additive group of rationals isomorphic to the multiplicative group of non-zero rationals.

2. Originally Posted by thomas_donald
Can I get help on the following problem:
Is the additive group of rationals isomorphic to the multiplicative group of non-zero rationals.
Assume that such an isomorphism does exist.

Let

$\displaystyle \theta$ : $\displaystyle (\mathbb{Q}, +)$ $\displaystyle \rightarrow$ $\displaystyle (\mathbb{Q} ^ \times,\times)$

Then for all $\displaystyle q \in \mathbb{Q}$ we have:

$\displaystyle \theta(q + q) = \theta (q) \theta(q)$

as $\displaystyle \theta$ is an isomorphism.

Can you see where a contradiction could lie in this arguement?

Spoiler:
As $\displaystyle \theta$ is an isomorphism, there exists a q such that $\displaystyle \theta(q) = 2$, so...

Hope this helps.

3. Thanks a lot. I get it.
If we take 2=(2q)=(q)*(q) we get: (q)=square root of 2
which does not belong to Q.
Great.

4. nice one.

5. Originally Posted by thomas_donald
Can I get help on the following problem:
Is the additive group of rationals isomorphic to the multiplicative group of non-zero rationals.
There is another way to see the non-isomorphism. In $\displaystyle \mathbb{Q}^{\times}$ there exists an element, different from the identity, which when multiplied on itself produces the identity. This is $\displaystyle -1$ since $\displaystyle (-1)^2=1$. However, in $\displaystyle \mathbb{Q}$ under addition you cannot find an element which when added to itself will produce the identity other than the identity. Thus, the two cannot be isomorphic.