Questions tagged [profinite-groups]
For questions regarding profinite groups and their properties.
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$H^2(G,A)$ is in bijection with the class of extensions of $A$ by $G$ - does this depend on the action of $G$ on $A$?
Let $G$ be a profinite group, and $A$ an Abelian group.
Given an extension of $A$ by $G$, $0 \longrightarrow A \longrightarrow E \longrightarrow G \longrightarrow 1$, it is known that we can make $A$...
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15 views
CoRes $\circ$ Res $ = [G:H]Id$ on the cohomology groups of a profinite group
Let $G$ be a profinite group, $H \leq G$ open. It is known that thus $H$ is closed and has finite index in $G$.
Any $G-$module is an $H-$module and one can construct the restriction map as the ...
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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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1answer
14 views
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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24 views
$p$-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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1answer
20 views
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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25 views
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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1answer
31 views
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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0answers
11 views
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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56 views
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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2answers
54 views
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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23 views
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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0answers
24 views
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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1answer
38 views
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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1answer
23 views
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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1answer
24 views
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 ...
2
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1answer
56 views
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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1answer
45 views
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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39 views
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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0answers
93 views
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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1answer
26 views
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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1answer
46 views
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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0answers
33 views
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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1answer
61 views
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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0answers
14 views
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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48 views
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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41 views
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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1answer
56 views
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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0answers
50 views
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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34 views
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\...
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52 views
Profinite groups as Galois groups over $\Bbb C(X)$
$\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\A}{\Bbb A}
\newcommand{\ab}{\mathrm{ab}}
\newcommand{\Gal}{\mathrm{...
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2answers
139 views
Is $\Bbb Z_p^2$ a Galois group over $\Bbb Q$?
$\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\A}{\Bbb A}
\newcommand{\ab}{\mathrm{ab}}
\newcommand{\Gal}{\mathrm{...
2
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1answer
75 views
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 ...
2
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1answer
72 views
Abelianization of free profinite group
Let $k$ be a cardinal (possibly infinite). Denote by $F_k$ the free group of rank $k$ and consider its profinite completion $G := \widehat{F_k}$.
What is the (topological) abelianization $G'_k := G ...
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0answers
59 views
Can $\mathrm{Aut}(\Bbb C)$ be turned into a profinite group?
It is known that for any (possibly infinite) Galois extension $L/K$, the Galois group is profinite, when endowed with the Krull topology (inherited from the inverse limit).
Can $\mathrm{Aut}(\Bbb C)...
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0answers
57 views
Dense abstract free subgroups in a free profinite group
I'm reading the third chapter of Profinite Groups by Ribes-Zalesskii. We can find many analogies between abstract free groups and free pro-$\mathcal{C}$ groups in the book- but I think I'm missing or ...
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1answer
81 views
The image of an Artin representation is finite
First let us define the necessary terms for the problem:
Definition
Let $G$ be a profinite group and let $k$ be a topological field. An $n$-dimensional representation of $G$ over $k$ is a ...
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Suitable sets, or set of generators converging to 1 in topological groups, mainly, profinite groups
What is the importance of set of generators converging to 1 in topological groups, and more specifficaly, in profinite groups? What is the motivation behind this definition?
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1answer
68 views
Every profinite group is naturally an affine group scheme over $\mathbb Q$?
I saw a special case of this after reading about the Weil group and the Weil-Deligne group. Since I'm trying to become more technically proficient in algebraic geometry, I thought this was ...
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1answer
49 views
Profinite completion of integers [closed]
Can someone help me in this exercise?
Show that $Z_{p}=\varprojlim_{n\in \mathbb{N}}\mathbb{Z}/p^{n}\mathbb{Z}$ can also be identified with the set of power series
$$Z_{p}= \{ b=\sum_{n=0}^{\infty}...
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27 views
Example Powerful Pro-$p$-Groups
I'm seeking for
some nice examples for powerful pro-p-groups* for prime $p \neq
2$.
By definition a powerful $p$- group $G$ is definined by following property:
The commutator $[G,G]$ is contained ...
3
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1answer
69 views
$H^1(G, \mathbb{Z}/p \mathbb{Z})$ and linearly independent elements in open subgroups.
Let $G$ be a profinite group and $p$ a prime number, and consider the following condition on $G$:
For every open normal subgroup $U$ of $G$ and any integer $N \geq 0,$ there are $N$ elements $$z_1, \...
3
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1answer
62 views
Conjugate subgroups in inverse limit of groups
Let $(G_i)_{i\in I}$ be a cofiltered system of finite groups and let $(H_i)_{i\in I}$ such that $H_i\subset G_i$ are subgroups and the $H_i$ form a compatible cofiltered system to the one of the $G_i$,...
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63 views
Question about why inertia subgroup is a closed subgroup of the Galois group
Let $K$ be a local field complete with discrete valuation $v$ with finite residue field $k$.
I am just wondering how does one know that the the inertia subgroup $I_v$ is a closed subgroup of $Gal(\...
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1answer
74 views
Infinite Cyclotomic Extension of $\mathbb{Q}$
Let $\mu_n$ be the group of solutions to $x^n - 1$ over $\mathbb{Q}$, and let $\Omega$ be the compositum of all $\mathbb{Q}(\mu_n)$ (in some fixed algebraic closure). I'm trying to understand why
$$
\...
4
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1answer
72 views
Relation between profinite abelian groups and $\hat{\Bbb{Z}}$ modules
Suppose we have a profinite abelian group $A$, then we can define a $\hat{\Bbb{Z}}$-module structure on $A$. However, not every $\hat{\Bbb{Z}}$-module is a profinite group, as one can see for example ...
3
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1answer
109 views
Profinite completions and inverse limits
The following questions are mainly curiosities; I recently came across the definitions of inverse limits and profinite completions, and below are two questions about them I could not answer myself. I ...
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0answers
42 views
$Ext(-,\mathbb Z)$ for profinite (Galois) groups
I start from the vague question "how should a Galois group behave"?
For example, what can be said for $Ext^1(G,\mathbb Z)$ when $G$ is a Galois group? And what about the behaviour of $Ext^1(G,A)$ for ...
0
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2answers
36 views
Expressing the inverse limit of a system of inverse limits of profinite spaces in a specific way
I am trying to prove something about free profinite modules, but the sticking point is entirely an issue about profinite spaces. Specifically, I need to express a profinite space $X$ in a very ...
