Let $G$ be a (countable) residually finite group whose profinite completion is topologically finitely generated. Must $G$ be finitely generated?



Finite generation and profinite completion
No. Take $G=\oplus\_{p} C\_p$, where the sum is over all primes $p$ and $C\_p$ is the cyclic groups of order $p$. Then $G$ is clearly locally finite and infinite and therefore not finitely generated. However, it is easy to see that the profinite completion of $G$ is $\Pi\_p C\_p$ which is a cyclic profinite group. 


