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:
Hope this helps.
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.