# Questions tagged [profinite-groups]

For questions regarding profinite groups and their properties.

172 questions
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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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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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 $$Z_{p}= \{ b=\sum_{n=0}^{\infty}... 0answers 27 views ### Example Powerful Pro-p-Groups I'm seeking for some nice examples for powerful pro-p-groups* for prime p \neq 2. By definition a powerful p- group G is definined by following property: The commutator [G,G] is contained ... 1answer 69 views ### H^1(G, \mathbb{Z}/p \mathbb{Z}) and linearly independent elements in open subgroups. Let G be a profinite group and p a prime number, and consider the following condition on G: For every open normal subgroup U of G and any integer N \geq 0, there are N elements$$z_1, \...
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 $G_i$,...