What is the profinite completion of the group $S^1$? 


where $S^1= \{ z\in\mathbb{C}: |z|=1 \}$ is a compact and abelian group.



Profinite completion
It is trivial, as $S^1$ has no non-trivial (normal) finite index subgroups. 


Recall that the profinite completion is the inverse limit of the $G/N$ where $N$ is a normal subgroup of finite index. So, if the only normal finite index subgroup is the full group, the profinit completion is trivial. And, for your group this is the case, for example as it is divisible, or by a direct argument. 


