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 ?

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    $\begingroup$ Closed subgroups of free profinite groups are exactly the projective profinite groups. $\endgroup$ Jan 3, 2017 at 17:19

1 Answer 1


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.


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