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 $f$ […]

Intereting Posts

Old AMM problem
Extensions and contractions of prime ideals under integral extensions
Is “connected, simply connected” Redundant?
Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{\sqrt{n}}$ converge?
n people & n hats: probability that at least 1 person has his own hat
Does the series $\sum_{n = 1}^{\infty}\left(2^{1/n} – 1\right)\,$ converge?
How will this equation imply PNT
Geometric derivation of the quadratic equation
Product of Two Metrizable Spaces
Exponential bound on norm of matrix exponential (of linear ODE)
How the product of two integrals is iterated integral? $\int\cdot \int = \iint$
Legendre polynomials, Laguerre polynomials: Basic concept
Addition is to Integration as Multiplication is to ______
Continuous deformations of points in $\mathbb{R}^n$ in a monotonic fashion
Motivation for different mathematics foundations