I am currently working through a problem on finite abelian groups.
Suppose we have distinct primes $p\neq q$ which both divide the order of a finite abelian group $G$. The first part asks to show that there is thus a cyclic subgroup of $G$ of order $pq$, which I have managed to do. The next part asks to prove that there is a quotient of $G$ which is cyclic and of order $pq$.
My guess is that if $H$ denotes our cyclic subgroup of order $pq$ in $G$ that the quotient should look something like $$\frac{G}{G/H}\simeq H$$ Where we have to find a way of identifying the quotient $G/H$ with a subgroup of $G$.
Is this the right direction to take things, or should the proof follow a different line of reasoning?