# Questions tagged [profinite-groups]

For questions regarding profinite groups and their properties.

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### Extend an action of indeterminates to an action of field of rational functions

In this text in the theorem 1.15 the author constructs a Galois extension using group action. In his construction, I can see why $G$ can be seen as an automorphism group of $K$, but I cannot see why ...
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### Showing torsion abelian groups have a unique $\widehat{\mathbb{Z}}$-module structure.

Let $E$ be a torsion abelian group, we write it additively. Define the multiplication \begin{align} \widehat{\mathbb{Z}} &\times E \rightarrow E, \\ (a,g) &\mapsto a\cdot g:=a_ng, \end{align} ...
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### Proving that the first Galois cohomology group is direct limit of finite quotients

This question comes from Silverman's Arithmetic of Elliptic Curves, specifically the appendix on Galois cohomology. I am a cohomology beginner, interested (for now) in understanding just enough to get ...
1 vote
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### Closed subgroups of profinite groups and basis of neighbourhoods

Let $G$ be a profinite group with a basis of neighbourhoods $U_n$ of normal subgroups. Furthermore let $H\subset G$ be a closed subgroup. Then we can define the open subgroup $H_n:= H\cdot U_n$. Is ...
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### A torsion-free abelian pro-$p$ group is free

I will give some context before statement my problem. Definition 1. A map $\theta: X \to G$ from a ser $X$ to a profinite group $G$ is said to be $1$-convergent if $\{x : \theta(x) \not\in N\}$ is ...
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1 vote
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### Restriction-Induction Lemma for representations of standard split maximal torus of $GL_2(F)$

I am trying to understand the proof of the following lemma, which comes from section 9.3 of The Local Langlands Conjecture for GL(2). In this question, $T$ is the subgroup of $GL_2(F),$ $F$ a non-...
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### Lemma 8.1 of "The Local Langlands Conjecture for GL(2)"

Let $F$ be a non-Archimedean local field, and $N$ the subgroup of $G=GL_2(F)$ of the form $\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}$ with $x \in F$. Let $(\pi,V)$ be a smooth ...
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1 vote
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### Given a smooth representation $(\pi,V)$ of a group $G$, what is the natural smooth representation of $G$ on $V/U$, where $U$ is $G$-stable.

I have the following problem: Let $(\pi,V)$ be a smooth complex representation of a locally profinite group $G$. Let $U$ be a $G$-subspace of $V$. Show that there is a natural representation of $G$ on ...
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### Deriving functors on subcategories and profinite group cohomology

I am reading through Weibel's chapter on Galois Cohomology and there he defines profinite group cohomology as the right derived functors of the $G$-invariants functor, but restricting to $C_G$, a ...
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1 vote
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### Every Field Extension is Composite of its finite subextensions

In Page 279 of Algebriac Number Theory by Neukirch, it states that every field extension is composite of its finite subextensions. I know how to prove this in Galois Theory, but I don't see how to ...
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### How to compute the unitary dual of a noncommutative profinite group?

Let p $\neq 2$ be a prime number. Let $G=\mathbb{H}(\mathbb{Z}_p)$ be the group of uni-triangular 3x3 matrices wih entries in the ring of p-adic integers, sometimes called the profinite three ...
1 vote
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### Why there exists some finite index normal subgroup of $G_K$ which fixes all elements of $M$?

Let $G_K$ be an absolute Galois group of algebraic field $K$. Let $M$ be an finite abelian group. Let $G$ acts continuously on $M$($G$ with krull topology and $M$ with discrete one), then I want to ...
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### The $p$-completion of a knot group

In Serre's LNM 5, in the section on profinite groups, he give the exercise: Let $k$ be a knot in $\mathbb{R}^3$, and let $G= \pi _1( \mathbb{R}^3-k)$ be the knot group of $k$. Show that the $p$-...
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### A profinite group that is not finite is not countable

I am trying to show the well-known fact that if a profinite group $G$ is not finite, then is not countable (i.e. $|G|\geq 2^{\aleph_0}$). There are many ways of proving this fact but I am interested ...
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### Definition of surjective map $d_K = 1/f_K d : G -> \widehat{\mathbb{Z}}$ in Neukirch 4.4 Abstract Valuation Theory
The setup of the question is the same As the unanswered question here. Here is the relevant part of Neukirch. Here $G$ is an arbitrary profinite group (but I take it as the Galois group of some ...