The question is already in the title.
It is known that any subgroup of a free group is free. My question is:


**Is a closed subgroup of a free profinite group is again a free profinite group ?** 



Subgroup of free profinite group is free profinite?
No. The free profinite group $\widehat{\mathbb{Z}}$ on one generator is the direct product of the groups $\mathbb{Z}\_p$, $p$ prime.
Therefore each $\mathbb{Z}\_p$ is a closed subgroup of $\widehat{\mathbb{Z}}$, but is not free as a profinite group.


