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Questions tagged [profinite-groups]

For questions regarding profinite groups and their properties.

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Profinite lie groups

I have been trying to find out examples of couple of profinite groups with certain properties. Firstly, I know that $GL_n(\mathbb{Z}_p)$ is not lie group but I suspect that $GL_n(\mathbb{Z}_p/p)$ ...
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Open normal subgroups of a profinite group form a basis for the open neighborhoods of 1

In the following link: An equivalent definition of the profinite group I'm having trouble understanding the following quote from the answer: So if you replace each subgroup in a basis by the ...
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Two versions of Baire Category Theorem for Profinite Groups

Let $G$ be a profinite group: Theorem 1. Let $C_{1},C_{2},...$ be a countably infinite set of nonempty closed subsets of $G$ having empty interior. Then $$G \neq \bigcup_{n}^{\infty}C_{i}.$$ ...
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Subset of a subgroup is not closed under group actions

In the following image (from "Field Arithmetic by Fried & Jarden" Page 6, Lemma 1.2.2(b)), red rectangle, I'm trying to figure out why it's right to claim $h^{-1} \in H$. I thought the ...
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A subset $C$ of a pro-finite group is closed iff $C$ is compact

I'm trying to prove both directions.. I think I managed to prove one: if $C$ is compact then $C$ is closed. Feel free to correct me: Every pro-finite group is compact, Hausdorff, and has a basis ...
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Inclusion and pullback sheaf

Let $X$ be a topological space and let $S \subset X$ be a subspace with induced topology (not necessarily open or closed). Let $i : S \to X$ be the inclusion map. Assume moreover that for any ...
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Galois group of $\overline{F}/F$

Let $K$ be the algebraic closure of $F$ where $F$ is a finite field. Show that $Gal(K/F) \simeq \hat{\mathbb{Z}}$. I know that $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}_{n}$, so its enough to show ...
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Isomorphism between $\mathrm{Gal}(L_{1}L_{2}/L_{2}) \to \mathrm{Gal}(L_{1}/L_{1} \cap L_{2})$ (infinite extensions).

Let $K/F$ be a field extension. If $L_{1}$ and $L_{2}$ are between fields, prove that if $L_{1}/F$ is Galois, then $L_{1}L_{2}/L_{2}$ is Galois and there is a natural isomorphism $$\mathrm{Gal}(L_{1}...
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Krull Topology on Galois Groups (two equivalent(?) definitions).

I'm starting to study Infinite Galois Theory and its relation with Profinite Groups, but I'm having troubles with basic definitions. Definition 1. Let $K/F$ a Galois extension. Write $$\mathcal{F}...
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The multiplicative group of units in $\mathbb{Z}_{p}$ is isomorphic to $\mathbb{Z}_{p} \times C_{n}$ with $n=\max\{p-1,2\}$.

Problem. (a) Write down explicitly the rules for addition and multiplication in $\mathbb{Z}_{p}$. (b) Show that $(\mathbb{Z}_{p^{i}},\varphi_{ij})$ where $\varphi:\mathbb{Z}_{p^{j}} \to \mathbb{...
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Inverse limit of projective profinite groups is projective

I'm trying to prove the following (H.W question): Let $ G $ be inverse limit of projective profinite groups. Prove that $G$ is projective group. Projective group means "cohomological dimension ...
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Elements conjugate in profinite completion

Problem. Let $G$ be a residually finite group, and identify $G$ with its image under the canonical map to its profinite completion $\hat{G}$. Let $x,y \in G$. Prove that the following conditions are ...
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If $H$ is a closed subgroup of a profinte group $G$, then $H$ is the inverse limit of the open subgroups of $G$ containing $H$.

Show that if $H$ is a closed subgroup of a profinte group $G$, then $H$ is the inverse limit of the open subgroups of $G$ containing $H$. I was able to show that $H$ is profinite. I use a ...
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Book-recommendations “Profinite Groups”

I started studying Profinite Groups a few weeks ago. I'm using the book "Profinite Groups" by Wilson as a basis, but the book is not clear enough sometimes (probably because I'm studying it for the ...
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How is the base of the topology on a profinite group?

If $G$ and $H$ are profinite groups and $\varphi: G \to H$ is a homomorphism, then $\varphi$ is continuous iff the inverse image of any open normal subgroup in $H$ is open in $G$. If $\varphi$ is ...
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If $(X_{i}, \varphi_{ij})$ is a inverse system of nonempty sets and surjective maps, then the inverse limit is nonempty (proof verification)

I had a troubles with this problem, so I thought it would be important to write my solution because I'm not sure of all the details. Since I already accepted the answer, I can not open a bounty to ...
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If $(X_{i}, \varphi_{ij})$ is a inverse system of nonempty sets and surjective maps, then the inverse limit is nonempty.

Problem. Prove that if $(X_{i}, \varphi_{ij})$ is a inverse system of nonempty sets and surjective maps indexed by a countable direct set, then the inverse limit is nonempty. I proved some similar ...
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Subgroup of ${\rm Aut}\,(\widehat{\mathbb{Z}})$

Ribes and Zalesskii Corollary 4.4.8 show that the group of continuous automorphisms of $\widehat{\mathbb{Z}}$ satisfies ${\rm Aut}\,(\widehat{\mathbb{Z}})\cong\mathbb{Z}_2\times\frac{\mathbb{Z}}{2\...
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Does the Burnside $\mathbb Q$-algebra $A$ of a group depend only on $\dim_{\mathbb Q}A$?

The Burnside $\mathbb Q$-algebra $\mathbb QB(G)$ of a group $G$ is usually considered only when $G$ is finite; see Section 3.1 of the text [1] Serge Bouc, https://pdfs.semanticscholar.org/aff3/...
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finiteness of an abelian topological group

Let A be an abelian (Hausdorff) topological group. Assume that (1) the set of its torsion elements, and (2) a finitely generated subgroup are dense subsets of A. My question: must A be finite? (...
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Direct proof that closed subgroups of profinite groups are profinite

Each of the references that I check prove first the relatively involved characterization of profinite groups as compact Hausdorff totally disconnected topological groups, and then appeal to the fact ...
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Is this proof that $\widehat{G_p}$ is pro-$p$ free correct?

Let $G$ be an abstract group with the following presentation: $$G \simeq \langle x,y \mid x^2y^2 = 1 \rangle $$ Let $p \neq 2$ be an odd prime. I want to show that $\widehat{G_p} \simeq \mathbb{Z}_p$...
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Why is $g \mapsto (gU)$ onto the profinite completion?

Let $\mathcal{C}$ denote a variety of finite groups (closed to subgroups, finite direct products, quotients). Let $G$ be a compact group, and assume there is a local basis $\mathcal{B}$ of normal ...
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$p$-completion is pro-$p$ free

Let $G$ be an abstract finitely generated residually finite group, and suppose that it's $p$-completion $\widehat{G_p}$ is a pro-$p$ free group. Does this implies that $G$ is a free group? The ...
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Extension of surface group by cyclic is residually finite

Let $G$ be the fundamental group of a surface, and consider an extension $1 \to \mathbb{Z}/p\mathbb{Z} \to E \to G \to 1$. Is $E$ residually finite? I'm interested in proving the injectivity of the ...
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Open subgroup of $Gal(k_s| k)$

Let $k$ be a field with fixed separable closure $k_s$. Consider the profinite group $Gal(k_s|k)=: G$ with the corresponding topology. Furthermore, let $U \subset G$ be an open subgroup. Every subgroup ...
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Bound on the Number of Normal Subgroups of Index $n$

I'm reading Tamas Szamuely's "Galois Groups and Fundamental Groups" and have a question about an argument used in lemma 3.4.11 on page 83: Here $\hat{F}(X)$ is a free profinite group of finite rank $...
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Puntual convergence of function in profinite integers

Consider the topological ring of profinite integers $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n!\mathbb{Z}$ where an element $r \in \hat{\mathbb{Z}}$ is a tuple $(r_n)_n$ such that $r_m = r_n \mod(n!)...
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What is the profinite completion of a free abelian group of infinite rank?

By definition, profinite completion of a group $G$ is $\widehat{G}=\varprojlim_N G/N$ where $N$ runs through every subgroup of finite index in $G$. Let $M=\bigoplus_{n\ge1} \Bbb{Z}$ be a free abelian ...
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Why is the Hausdorff property mentioned in the characterization of profinite groups?

Profinite groups are usually characterized as compact, totally disconnected, Hausdorff groups. However, as shown here, every totally disconnected topological group already has the Hausdorff property. ...
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A certain colimit of representables sheaves, namely group actions, is a sheaf: why?

In the first answer to this post on MO, one finds that When you look at the category of sheaves on the category of finite action with the natural topology (covering are surjection of finite ...
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Direct limit of $L^2$ spaces

Let $G$ be a discrete group and consider all the subgroups $N$ of finite index, these form a directed as follows: if $N \subseteq N'$ (in this case we say that $N' < N$) there is a morphism $i_{N',...
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Index of a common normal core

Let $A, B, C$ be infinite groups and suppose that there are injective homomorphisms $\iota_A \colon C \to A$ and $\iota_B \colon C \to B$ such that $|A : \iota_A(C)|, |B: \iota_B(C)| < \infty$. ...
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Profinite group, and why the product of subgroups is closed

The following lemma is taken from Wilson's book Profinite Groups; Let $( H_i : i \in I)$ be a family of normal subgroups of a profinite group $G$. Assume $G = \overline{< \cup_{i \in I} ...
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If $G$ is profinite, and $A$ is discrete, $f: G \to A$ is continuous $\implies$ $f$ factors through a normal open subgroup

Let $G$ be a profinite group; that is compact, and totally disconnected. Take $A$ a discrete space, and a continuous map $f: G \to A$. $\exists N$ open and normal in $G$ and a continuous map $g: G/...
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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 ...
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Every group of totally disconnected type is locally profinite?

Let $G$ be a Hausdorff topological group in which every point has a neighborhood basis of open compact neighborhoods. Let's call this a group of totally disconnected (td)-type. On the other hand, we ...
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Is every totally disconnected topological group locally profinite?

Let $G$ be a topological group which is totally disconnected. Then one point sets in $G$ are closed, and hence $G$ is Hausdorff. On the other hand, we have a notion of a locally profinite group, a ...
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group homomorphism from a profinite group continuous iff kernel open

I have a question regarding a (probably simple) fact. However I am lacking some basic topological knowledge. Let $G$ be a locally pro finite group, i.e. ever open neighborhood of $1_G$ contains a ...
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Index of congruence subgroups in $GL_2(\mathbb{Z}_p)$ modulo their centers.

Let $\Gamma_i$ be the set of matrices in $GL_2(\mathbb{Z}_p)$ which are congruent to $1$ modulo $p^i$, that is they are the congruence subgroups. I know that $\Gamma_i$ is a pro-$p$ group and $\...
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Dualizing module and finiteness hypothesis

Serre, in his Galois Cohomology, states: Proposition 17. Let $n$ be an integer $\geq 0$. Assume: (a) $\text{cd}(G) \leq n$ (b) For every $A \in C^f_G$, the group $H^n(G, A)$ is finite. ...
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Do dense topological subgroups always have the same finite quotients as their underlying bigger topological subgroup?

Definition: Let $G$ be a topological group. We call $G$ a finite quotient of $G$ if there exists a normal subgroup $N$ of $G$ such that $H = G/N$ and $H$ is finite. Let $G'$ be a dense subgroup of $G$...
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Topology topics for Profinite Groups

I want to start studying Profinite Groups, but I have not yet done a formal course about Topology. I have already studied: continuity, open, closed, compact, connected and not much beyond that. I'd ...
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Show that $\mathbb{Z}[\sqrt{-5}]_3 \simeq \mathbb{Z}_3[\sqrt{-5}]$

I have a somewhat tedious question. Is $\mathbb{Z}[\sqrt{-5}]_3 \simeq \mathbb{Z}_3[\sqrt{-5}]$ ? It might help to describe what these are: $\mathbb{Z}[\sqrt{-5}]_3$ is the $3$-adic completion of ...
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Virtual finite cohomology implies finite cohomology

Let $G$ be a torsion-free pro-$p$ group, and let $H$ be an open subgroup of $G$. Suppose both $H$ and $G$ have finite cohomological dimension. What I want to show is: If all groups $H^i(H, \mathbb{...
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Are we allowed to interchange product and inverse limits?

Currently, I am trying to show that the profinite completion $\hat{\mathbb{Z}}$ of $\mathbb{Z}$ is isomorphic to $\prod_p \mathbb{Z}_p$ (as topological groups) where $p$ runs through all prime numbers ...
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Is $\mathbb{Z}_p[\omega]$ profinite when $\omega$ is a primitive $p$-root of unity?

Let $\mathbb{Z}_p$ be the $p$-adic integers. We have that the $p$-th cyclotomic polynomial is irreducible over $\mathbb{Z}_p$ applying the Eisenstein criterion (which is valid over $\mathbb{Z}_p$ when ...
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Show that G is profinite

Let G be a compact group, H be an open subgroup of G. Show that if H is profinite, then G is also profinite. Lemma to use as a hint is this: Let G be a compact group and $ {N_i | i \in I}$ be ...
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Split extensions of $G/U$ by elementary abelian $p$-group and $H^1(U, \mathbb{Z}/p\mathbb{Z})$

This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 \to P \to E \to W \to 1$ a finite group extension and $f\colon G \to W$ a continuous ...
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Upper central series, semidirect products and a lifting property (argument verification)

Let $G$ be a profinite group, and define the lifting property: $(*_p)$ For every extension $1 \to P \to E \to W \to 1$ where $E$ is finite and $P$ is a $p$-group and for every surjective morphism $f\...