Prove or disprove if its isomorphic.

Question

$(Q^+,*)$ is isomorphic to its subgroup, consisting of rationals $\frac p q$ where $p$ and $q$ are odd integers.

I don't know about the cardinality of an infinite set. If two groups are isomorphic then they have the same cardinality. So how do I define a one-to-one map between these two groups? My assumption is that isomorphism between these two groups do exist, Since I couldn't find any counter-example. Please explain in detail since I am new to the group theory.

Answer 1

Hint: Let $2=p_1,p_2,p_3,\dots$ be the list of primes, and extend the mapping $p_n\mapsto p_{n+1}$.