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 ...
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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$. Is ...
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$SL_n(\mathbb{Z}_p)$ as inverse limit

I'm studying a proof that $SL_n(\mathbb{Z}_p)$ is profinite, but I'm stuck in one point. If $a$ is a matrix, then $a^{(i,j)}$ denotes the entries of $a$. The idea is show that $$SL_n(\mathbb{Z}_p) \...
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Morphism of inverse systems

All maps in the lemma hypothesis are continuous. This is a lemma from Ribes's Profinite Groups. In this proof there is a claim that I can't see why is true. Lets suppose that all topological spaces ...
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$G$-composition length of representation

I do not understand what is meant by $G$-composition length. The textbook I'm using - The Local Langlands for GL(2) - makes the following statement. Let $\chi=\chi_1 \otimes \chi_2$ be a character of ...
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Groups with "same" finite-index normal subgroups

If $G, G'$ are two groups whose categories of finite-indexed normal subgroups are equivalent. Then, are they or their profinite completions isomorphic ? If instead G, G' are topological groups (...
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Krull topology is discrete if only if the exntension is finite

Let $M/K$ be arbitrary extension. Then, we can define Krull topology on $Gal(M/K)$ by taking {$Gal(M/F)|K⊆F⊆M,[F:K]<∞$} as fundamental system of neighborhood of $0$. I guess $Gal(M/K)$ has discrete ...
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Profinite Cohomology of $\hat{\mathbb{Z}}$: Abstract proof using $\delta$-functors

Lets $G=\hat{\mathbb{Z}}$ be the profinite completion of the integers, let $T$ be the topological generator of $G$. I'm interested in proving $$H^i(G,M)=\begin{cases} M^G & i= 0\\ M/(T-1)M &i=...
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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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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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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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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 ...
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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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Galois group of $\mathbb{Q}(\underset{n\geq 1}{\bigcup}\mu_n)/\mathbb{Q}$

I was trying to formalise the fact that $G_{\mu_\infty}:=Gal(\mathbb{Q}(\underset{n\geq 1}{\bigcup}\mu_n)/\mathbb{Q})\simeq \underset{n}{\varprojlim}\ (\mathbb{Z}/n\mathbb{Z})^{\times}=\hat{\mathbb{Z}}...
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Profinite completion of integers, $\hat{\mathbb{Z}}$, is isomorphic to the product over primes of the $p$-adic integers $\mathbb{Z}_p$

I've been trying to show that the profinite completion of the integers, $\hat{\mathbb{Z}}$ is isomorphic to the product over $p$ of the $p$-adic integers. But I'm kind of stuck. Here is what I got so ...
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Topologies on $Gal( \bar{\mathbb{Q}}/\mathbb{Q})$

Let $G_{\mathbb{Q}}:=Gal( \bar{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. We have that $$G_{\mathbb{Q}}\simeq \underset{K}{\varprojlim}\ Gal(K/\mathbb{Q})\subset \underset{...
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Proofs with strong approximation theorem

I am stuck on some proofs concerning strong approximation in Chapter 3.1 of Hida's book on modular forms. I have put in green the things that I do not understand. The set $gL\subset \mathbb{A}^\infty$...
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Equivalent definition of profinite group

Sorry for my bad English. Let $G$ be a topological group, then we say $G$ is profinite if $G$ satisfies next equivalent condition as follow; (i) $G$ is compact, Hausdorff, totally disconnected. (ii)$G$...
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Existence of deep enough open subgroups in profinite groups

Let $G$ be an infinite profinite group and $\{U_i\}_{i \in I}$ be any family of open subgroups of $G$. Is it possible to choose a family of open subgroups $V_i < U_i$ with the property that, for ...
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1 answer
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Intermediate subgroup between closed subgroups of profinite group is closed?

Let $G$ be a profinite subgroup. And $K ,H$ be closed subgroups of $G$ such that $K \vartriangleleft H$ and $(H:K) < \infty $ . $K \subseteq B \subseteq H$ (subgroups) Then $B$ is closed in $G$? ...
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Is it true that in a profinite group any set of generators has a subset of generators converging to 1.

$\underline{\text{Definition:}}$ We say that a subset $X$ of a profinite group $G$ converges to 1 if every open (normal) subgroup $U$ of $G$ contains all but a finite number of the elements in $X$. It ...
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Inverse limit and crossed product

Let $p$ be a prime and $H$ be a uniform, pro-$p$ group. Then the Iwasawa algebra $\mathbb{F}_p[[H]]$ can be seen as the $I$-adic completion of the group algebra $\mathbb{F}_p[H]$ for $I$ the ...
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Are there "simple" motivations for the profinite topology?

I'm taking a topology course right now, and a lot of people are (understandably) confused by the profinite topology (and pro-$\mathcal{C}$ topologies more generally) on a group. The definition is a ...
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Natural epimorphism from the closure to its profinite completion

Let $G$ be a finitely generated residually finite group. Let $H \le G$. We have pro topology on $H$ determined by profinite topology of $G$. I suppose this is same as subspace topology on $H$. Since $...
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Profinite completion of the $p$-adic integers $\mathbb Z_p$

$ \DeclareMathOperator{\bN}{\mathbb N} \DeclareMathOperator{\bZ}{\mathbb Z} $Just playing around with profinite completions. As a basic example (ignoring the trivial case of finite groups or $\bZ$) I ...
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Rank of a profinite group

in the book Field Arithmetic by Fried and Jarden, two different definitions are given for the rank of a profinite group. The first, in chapter 16, is for finitely generated profinite groups, and this ...
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Exercise about set of generators converging to 1 of a profinite group $G$.

I am trying to solve the following exercise. Let $X$ be a set of generators converging to 1 of a profinite group $G$. Then the topology on $X - \{1\}$ induced from G is the discrete topology. If $X$ ...
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Pro-$p$-group with countably many generators.

How can we show that a pro-$p$-group G with countably many generators can be defined by a projectiv system of finite $p$-group H_n, $n\in\mathbb{N}$ such that: _ $H_0=1$, _ surjective morphisms $\phi_{...
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On Frobenius reciprocity for (compact) induction of smooth representations of locally profinite groups

Let $G$ be a locally profinite group and let $H$ be a closed subgroup of $G$. Let $\sigma$ be a smooth (complex) representation of $H$. Then, we have two ways of producing a smooth representation of $...
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Example of nilpotent profinite group

Is there any example of a topologically finitely generated profinite group $G$ which is nilpotent as a group but has an infinite torsion-part? The question is motivated by the fact that an algebraic ...
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Smooth representation of profinite group as a limit

Let $G$ be a profinite group. As is well-known, we may write $G = \operatorname{lim} G/N$ for $N$ family of normal open compact subgroups of $G$ (which is a cofiltered limit, and we may index it by $\...
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1 vote
1 answer
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p-adic Analogue of Tannaka–Krein

The Tannaka-Krein duality gives us that a necessary and sufficient and necessary condition for a $\mathbb{C}$-linear monoidal category to be equivalent to the category of continous, finite dimensional ...
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1 answer
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Action of pro finite group is continuous? [closed]

Let $G$ be a pro finite group, and $E$ be discrete topological space. I heard that action $G×E→E$ is continuous. Could you tell me the proof of this statement? Reference is also appreciated. Thank ...
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Confusion about the Galois group $\mbox{Gal}(\mathbb{Q}(\zeta_{p^\infty})/\mathbb{Q})\cong \mathbb{Z}_p$

Let $p$ be a prime and $\zeta_{p^r}$ be a primitive $p^r$-th root of unity. Then many textbooks, e.g. Milner's, ask one to prove that $\mbox{Gal}(\mathbb{Q}(\zeta_{p^\infty})/\mathbb{Q})\cong \mathbb{...
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Cardinality of a elementary abelian pro-$p$ group

Let $G$ be an elementary abelian pro-$p$ group. Then we have that $$G=\prod\limits_{\mathfrak{m}}C_p$$ where $\mathfrak{m}$ is a cardinal. We have this as a direct consequence of Theorem 4.3.8. from ...
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2 votes
1 answer
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Irreducible subquotients of compact induction

Let $G$ be a locally profinite group and let $K$ be a compact open subgroup. All the representations are assumed to be smooth and complex. Let $\sigma$ be an irreducible representation of $K$. Is it ...
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1 vote
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Topological generator of profinite group

I'm trying to prove that $\pi(\operatorname{Spec}\mathbb{Z}[\frac{1}{2}])$ is topologically generated by its $2$-Sylows (Exercise 6.30 of Lenstra's notes on Galois Theory for Schemes). I've been given ...
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Describing the depth zero compact induction of a representation which is not cuspidal

Let $G$ be a reductive $p$-adic group. I would like to describe a representation of the type $$\rho = \mathrm{c-Ind}_K^G\,\sigma$$ where $K$ is a maximal parahoric subgroup of $G$ and $\sigma$ is an ...
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Are p-Sylow subgroups of a profinite group dense?

I'm trying to show that $\pi(\operatorname{Spec}\mathbb{Z}\left[\frac{1}{2}\right])$ is topologically generated by its 2-Sylow. I already know that the fundamental group of $\operatorname{Spec}\mathbb{...
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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 ...
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1 answer
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Profinite completion of locally profinite topological group

I was trying to prove the following lemma with two parts. Let $G$ be a locally profinite topological group. Denote the profinite completion of $G$ by $\hat{G} = \underset{N}{\varprojlim} G/N$, where $...
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Pronilpotent group that is not pro-$p$

Pronilpotent groups are defined as inverse limit of an inverse system of finite nilpotent groups. They generalize the class of pro-$p$ groups. I tried to find an example of a pronilpotent group that ...
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1 vote
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Distinct topologies giving same topologies

We have profinite group $C_p^{\mathbb{Z}}$ which is infinite countable cartesian product of finite cyclic group of prime order $p$. I understand this part that "The profinite group $C_p^{\mathbb{...
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