Recently I am reading Nathan Carter's Visual Group Theory. I saw this exercise:
Exercise 8.40. Recall the group $\mathbb Q$ (under addition) and the group $\mathbb Q^*$ (under multiplication) introduced in Exercise 4.33. Show that $\mathbb Q\times C_2 \cong \mathbb Q^*$ by specifying the isomorphism, and explaining why the function you give is indeed an isomorphism.
Here $\mathbb Q^*$ is the set of non-zero rational numbers and $C_2$ is the cyclic group of order 2.
But I think that I am not able to construct the isomorphism.
If the $\mathbb Q$ is replaced by $\mathbb R$ instead, then it is much easier:
Denote $C_2$ as $(\{1,-1\},\times)$,
one of the isomorphism is $f(a,b)=be^a$.
But now we are dealing with $\mathbb Q$, and most number to a rational number power is not rational (whatever base chosen).
So here's my question: am I on the right track? Is this done by exponential and I have missed something? Or it is nothing about exponential but actually something else?
Thanks in advance!