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.

Let $G$ be any group, and $\widehat{G}$ its profinite completion. Is it true that $\widehat{\widehat{G}}=\widehat{G}$, i.e. is it true that $\widehat{G}$ is (canonically isomorphic to) its own profinite completion? It seems that it should follow from the universal property of the profinite completion, but I don’t see how. Thanks in advance for any solutions or […]

Fix a prime $p$. Let $G$ be a group endowed with the pro-$p$ topology, and let $H$ be an open subgroup of $G$. I am trying to prove that the induced topology on $H$ is the pro-$p$ topology of $H$. It is enough to show that each normal subgroup $N$ of $H$ with $[H:N]$ a […]

Let $\mathsf{ProFinGrp}$ be the category of profinite groups (with continuous homomorphisms). This is equivalent to the Pro-category of $\mathsf{FinGrp}$. Notice that $\widehat{\mathbb{Z}} = \lim_{n>0} \mathbb{Z}/n\mathbb{Z}$ is a cogroup object, since it represents the forgetful functor $U : \mathsf{ProFinGrp} \to \mathsf{Set}$. Although $\mathsf{ProFinGrp}$ has no coproducts (?), the coproduct $\widehat{\mathbb{Z}} \sqcup \widehat{\mathbb{Z}}$ exists, it coincides by […]

Let $G$ be a compact group. A representative function $f\in\mathcal{C}(G,\mathbb{K})$ is a function such that $\dim\left(\operatorname{span}\left(Gf\right)\right)< \infty$. Remark that the representative functions form a subalgebra of $\mathcal{C}(G,\mathbb{K})$. I’m following the book “The Structure of Compact Groups” by Hofmann&Morris on this subject. I would like to be able to show that that a representative function $f$ […]

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