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Questions tagged [profinite-groups]

For questions regarding profinite groups and their properties.

Profinite groups are topological groups that can be "assembled" from finite groups. Precisely, a profinite group is a Hausdorff, compact, and totally disconnected topological group.

If given the discrete topology, every finite group is profinite, and the Galois theory of infinite degree field extensions gives rise to profinite Galois groups. Products and closed subgroups of profinite groups are also profinite.

Reference: Profinite group.

348 questions
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Why is the image of a group dense in its profinite completion?

I'm new to profinite groups and wish to prove the following, which is a claim I found in many sources, but it is probably so simple that nobody even proves it. Let $G$ be a group and $\hat{G}$ be its profinite completion. We have a canonical…
Shoutre
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Is the commutator subgroup of a profinite group closed?

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.
user93417
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Profinite group isomorphic to its completion

Here is a very simple claim: if $ G $ is profinite group isomorphic to its completion $ \hat{G} $. Then the natural map $ \phi: G \to \hat{G} $ is an isomorphism. I'm not sure how to show this. The universal property of ($ \hat{G}, \phi $) seems to…
user564167
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Continuous Image of a profinite group into $T=\mathbb{R}/\mathbb{Z}$

Let $f : G \to T$ be a continuous homomorphism from a profinite group into $T= \mathbb{R}/\mathbb{Z}$. According to a book I am reading, $f(G)$ is totally disconnected. Why is this true? The book doesn't explain it at all.
user417289
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Question about Neukirch's book Cohomology of Number Fields

Let $G$ be a profinite group. $A$ and $B$ are discrete G-modules. If $A^U=A$ for some open subgroup $U \subseteq G,$ then why Hom(A,B) is a discrete G-module. $\big( g(\phi)(a) = g(\phi(g^{-1}(a))) \big)$
math
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Inverse Limit of inverse surjective System

Let $\{G_i, \phi_{i,j}, i, j \in I\}$ (with a partially ordered $I$) be an inverse system of finite groups and let $G = \varprojlim G_i$ be its inverse limit, that is, a profinite group. If I know that all $\phi_{i,j}$ are surjective how can I…
user267839
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Definition of Profinite Groups.

We know that totally disconnected topological groups are Hausdorff. Then why we are considering both Hausdorff and totally disconnected conditions in the definition of profinite groups. Topological Group $G$ totally disconnected $\Rightarrow$ $G$…
math
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Inverse limit of an inverse system

A direct set is a partially ordered set $(I,\leq)$ s.t. $\forall a,b\in I,\;\;\exists c\in I$ with $a\le c$ and $b\le c$. An inverse system is a family of topological spaces indexed by $I$, $\{X_i\}_{i\in I}$ together with a family of continous maps…
Joe
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Convergence in $\hat{\mathbb{Z}}$

The following is part of an exercise from Lenstra's Galois theory for schemes.. Let $a=\frac{b}{c}\in \mathbf{Q}^\times$, $n\in \widehat{\mathbf{Z}}^\times$. Prove that there exists a sequence of positive integers $(n_i)_{i\geq 0}$ that satisfies…
Dr. Heinz Doofenshmirtz
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Extending a function to the profinite completion of the integers

I am trying to see if the function $f: \mathbb{Z} \to \mathbb{C}$ defined by $$ f(n) = \frac{1}{n-z} $$ for some $z \in \mathbb{C}\backslash \mathbb{R}$ can be continuosly extended to the profinite completion of $\mathbb{Z}$, denoted by $\hat…
k76u4vkweek547v7
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Map on Profinite Groups defined as Limes

Let $G$ be a profinite group and $ \bar{\mathbb{Z}} = \varprojlim _{n \in \mathbb{N}} \mathbb{Z}/n\mathbb{Z} \subset \prod _{n \in \mathbb{N}} \mathbb{Z}/n\mathbb{Z}$. Let $g \in G$ and define $a := (a_n + n\mathbb{Z})_{n \in \mathbb{N}} \in…
user267839
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Homomorphism onto a profinite group

Let $f:S^1\rightarrow G$ be a continuous homomorphism from the torus onto a profinite group G. Is it true that $f$ must be trivial? Note that if G is finite then this is true since the kernel is an open subgroup of $S^1$ of finite index (and…
Yanko
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Example of non-computable implicit operations

I used notation of paper of Jorge Almeida "Dynamics of implicit operations and tameness of pseudovarities of group" in my question. Let $V$ be a pseudovarity of finite groups. We know that every element $\pi$ of a $A$-generated free pro-$V$ group…
user182085
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Second countable profinite space

The Corollary 1.1.13 in my text book is the following statement. $\textbf{Corollary 1.1.13}$ A profinite space $X$ is second countable if and only if $$X\cong \varprojlim_{i\in I}X_i$$ where $(I,\leq)$ is a countable totally ordered set and each…
Combalge
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