# Subgroup of free profinite group is free profinite?

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 ?

• Closed subgroups of free profinite groups are exactly the projective profinite groups. Jan 3, 2017 at 17:19

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.