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Let $G$ be a profinite group, $[G,G]=\{ghg^{-1}h^{-1}|g,h\in G\}$ is a subgroup of $G$. Is $[G,G]$ closed?

In the case we are interested, $G$ is the absolute galois group of a local field.

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    I'm late but anyway, I think you meant the subgroup generated by the commutators $ghg^{-1}h^{-1}$ – Shoutre Apr 24 '18 at 23:00

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In general is not true that for a profinite group $G$ the derived subgroup $G'$ is closed; But if $G$ is a pro-$p$-group, with $p$ a prime, and if $G$ is finitely generated then it becomes true. For example this is true for the group of $p$-adic integer $\mathbb{Z}_{p}$.

The Number Theorist
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