TITLE: Show that $Q^+$ is not isomorphic to $Q^* $.
QUESTION [5 upvotes]: 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 $Q^*$.

REPLY [5 votes]: 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.