Show that $Q^+$ is not isomorphic to $Q^* $.

Question

Let $Q^+$ be the group of positive rational numbers under multiplication.

Let $Q^*$ be the group of all the rational number of the form $\frac{p}{q}$, where both $p$ and $q$ are odd positive integers, under multiplication.

Show that $Q^+$ is not isomorphic to $Q^*$.

Answer 1

If $p_1,p_2,\ldots=2,3,\ldots$ is the sequence of primes then the $\phi$ $$\begin{eqnarray} \phi(1)&=&1\\ \phi({p_i})&=&p_{i+1},\, \forall i\geq 1 \end{eqnarray}$$ can be extended to an isomorphism from $Q^+$ to $Q^*$. So they are isomorphic.