# Questions tagged [profinite-groups]

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### Dual of smooth induced representation

Let $G$ be a locally profinite group with a closed subgroup $H$ and a smooth representation $(\pi,V)$ . Denote by $Ind_H^{\infty,G}(\pi)$ the smooth induced representation of $\pi$.
Is there a nice ...

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158 views

### Finitely generated commutative rings with the same profinite completion

Let $R_1$ and $R_2$ be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: $\widehat{R_1}\cong \widehat{R_2}$.
Suppose that $R_1$ is a domain. Does ...

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46 views

### Is Pontryagin dual of an abelian torsion group strongly complete?

Let $A$ be an ablian torsion group. Pontryagin dual of $A$ is Hom$(A,\mathbb{Q}/\mathbb{Z})$. Is it strongly complete, i.e. every subgroup of finite index is open?

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75 views

### Commutator subgroup of the absolute Galois group - a closed subgroup

Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible that the commutator subgroup of the absolute Galois group of $K$ (considered as an abstract group) is a closed subgroup? This property ...

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46 views

### Homomorphism defined on a dense subset extending in different way with respect to different target topologies

Let $G$ be a profinite group with a dense subset $S\subset G$. Let $H_1$ and $H_2$ be two topological groups whose underlying abstract group is the same. Can there exist homomorphisms of topological ...

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103 views

### How a profinite group can be obtained from its normal open subgroups?

Let $\Delta$ be a set, each element of which is a profinite group (2 distinct elements of $\Delta$ may be isomorphic). Under what conditions on $\Delta$, there exists a profinite group $G$ which has $\...

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181 views

### Is there a free profinite abelian group on a profinite set?

Let $\mathit{Profinite}_{\mathrm{Ab}}$ be the category of profinite abelian groups, and let $\mathit{Profinite}_{\mathrm{Set}}$ be the category of profinite sets. Does the forgetful functor
$$\mathit{...

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208 views

### Measure of subsets of profinite groups

Let $G$ be an infinite profinite group, so $$G=\lim_{\longleftarrow}G/N$$ where $N$ runs through the open normal subgroups. I have two questions:
Is $G$ of Haar measure zero in the compact group $\...

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186 views

### Continuous function defined by measurable sets

Is the following slightly generalization of Corollary 20.17 in Hewitt and Ross Book (page 296) correct?
Let $A$ be a subset of a profinite group $G$ ( compact, Hausdorff, totally disconnected ...

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227 views

### Mackey theory in the setting of locally profinite groups

$\DeclareMathOperator\Hom{Hom}$Let $R$ be a commutative ring (not necessarily unital). Let $G$ be a finite group, and let $H_1, H_2$ be subgroups of $G$.
Recall the following standard result [1, Thm. ...

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591 views

### Does a (nice) centerless group always have a centerless profinite completion?

This is an extension of a question I asked here on Math.SE
Assume that I have a finitely generated residually finite centerless group $G$. Is it true that the profinite completion $\hat{G}$ also has ...

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262 views

### Meaning of epimorphism from full Galois group to some group

My problem has two parts: let $\;G:=\operatorname{Gal}(\overline{\Bbb Q}/\Bbb Q)\;$ be the full Galois group of the rationals and $\;K\;$ be some finite group, then:
(1) Does having an epimorphism (...

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179 views

### Applications of one of Serre's Theorems

This theorem is due to Serre:
Let $G$ be a profinite group, $p$ prime. Assume that $G$ has no
element of order $p$ and let $H \leq G$ be an open subgroup. Then
$cd_p(G) = cd_p(H)$.
Where $cd_p(...

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76 views

### Splitting the canonical projection to the free pro-p group

Let $\widehat F(k)$ be the free profinite group on $k$ generators and let $p$ be a prime. Then there is a canonical projection $\pi\colon \widehat F(k)\to \widehat F_p(k)$ where $\widehat F_p(k)$ is ...

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223 views

### Colimit of continuous cohomology over subgroups

Suppose $G$ is a profinite group, in fact in the applications I'm interested in it would be a $p$-adic analytic group similar to $GL_{n}(\mathbb{Z}_{p})$. Say $M$ is a profinite $G$-representation, ...

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110 views

### Are double cosets of cyclic subgroups separable in a special linear group?

Let $A,B \in \mathrm{SL}_3(\mathbb{Z})$. Set
$$S = \langle A \rangle \cdot \langle B \rangle = \{A^mB^n : m,n \in \mathbb{Z}\}.$$
Is $S$ closed in the profinite topology on
$\mathrm{SL}_3(\mathbb{...

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123 views

### For a pro-p, profinite group, abelianization being finitely generated is the same as being topologically finitely generated

I remember reading (without proof) that for $\Gamma$ a profinite, pro-$p$ group, the following are equivalent:
1) Every open subgroup $\Gamma_0$ is topologically finitely generated.
2) The ...

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193 views

### Linear representation of the free metabelian / 2-step nilpotent profinite groups on 2 generators

Let G be the free profinite group on 2 generators, $A=G/[G,[G,G]],B=G/[[G,G],[G,G]]$, then what is the structure of the groups $A$ and $B$?
I heard that $A$ is isomorphic to the group of such ($3\...

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216 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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158 views

### Duality between coalgebras and (pseudocompact) algebras - uniqueness

The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below.
Let $k$ be a field. The ...

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348 views

### Continuous cohomology of a profinite group is not a delta functor

Let $G$ be a profinite group, then there is a general notion of continuous cohomology groups $H^n_{\text{cont}}(G, M)$ for any topological $G$-module $M$ (I require topological $G$-modules to be ...

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301 views

### Profinite completion of finitely presented groups

Let $G$ be a finitely presented group, $\widehat{G}$ be the profinite completion of $G$, and $f: G\rightarrow \widehat{G}$ be the natural map.
My question is:
Is there an example of $G$ for which $\...

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93 views

### Profinite groups with finite torsion

Let $G$ be a profinite abelian group such that for every $x\in G$ and every $n\in\mathbb Z$ the preimage of $x$ under the multiplication by $n$ map is finite. Does it follow that the torsion subgroup ...

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136 views

### Dense abstract free subgroups in a free profinite group

Let $\langle a, b \rangle = F_2$ be a two-generator free group and $\hat{F_2}$ be its profinite completion. Is there an element $c\in \hat{F_2}$ such that $\langle a, b, c\rangle \le \hat{F_2}$ is ...

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259 views

### What are the LCA groups that are the Pontryagin dual of a locally profinite abelian group?

For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian):
finite groups $\leftrightarrow$ ...

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142 views

### maximal pro-l-quotients of absolute Galois groups

Let $K$ be a field, preferably a function field of a variety $X$ over $\overline{\mathbb{F}}_p$. I am looking for an answer or existing literature on the following question:
What is known about the ...

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195 views

### Torsion in a tensor product over a group ring

Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra.
Is it true ...

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### Maximal subgroups of infinite index and profinite completion

Preliminary remark: I'm mainly interested in an answer (or link to ressources) in the specific context of the first Grigorchuk group, but I believe that it may be of some interest to state the ...

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### Profinite closure of characteristic subgroup

Let $F$ be a free group of finite rank, and $K\subset F$ a finite index characteristic subgroup.
Let $\hat{F}$ be the profinite completion of $F$ (i.e. a free profinite group of same rank), and $\bar{...

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104 views

### Freeness of a quotient group

Let $p$ be a prime. Let $G=\langle x,y\rangle^{\textrm{pro-}p}$ be the pro-$p$ completion of the free group $\langle x,y\rangle$ generated by symbols $x$ and $y$. Define $G_{n+1}=[G,G_n]$ and $G_1=G$ (...

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180 views

### A question about continuous group cohomology

Let $G$ be a profinite topological group, $M$ a discrete $G$-module.
If $M$ is "P", is every $H^i_{\rm cont}(G,M)$ also "P"? or at least is it a subgroup/subquotient of an abelian group that is "P"? ...

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154 views

### Is there an elementary reason for why $SL_2(\mathbb{F}_p)$ for $p>5$ does not embed into $SL_2(\mathbb{Z}_p[w])?$

This is an exercise from Serre's book on Galois cohomology.
Let $p>5$ and consider the groups $SL_2(\mathbb{F}_p)$ and $SL_2(\mathbb{Z}_p[w])$ where $w$ is a primitive $p$th root of unity.
Is ...

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188 views

### No lifts in an exact sequence of profinite groups?

In pg. 24 of his book on Galois cohomology, Serre gives the following exercise:
"Give an example of an extension $1 \to P \to E \to G \to 1$ of profinite groups with the following properties:
(i) $...

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240 views

### On an inequality concerning the strict cohomological dimension of a profinite group

This is an exercise from Serre’s book on Galois cohomology.
Let $G$ be a profinite group and $H$ a normal closed subgroup and suppose that the cohomological dimension at the prime $p$ of $G/H$ is ...

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240 views

### The Unit Group of $\mathbb{Z}_p$

Let $\mathbb{Z}_p$ the ring of $p$-adic numbers. It's known that the multiplicative unit group $\mathbb{Z}_p ^\times$ can be set theoretically described as $\bigcup _{1 \le a \le p-1} a+ p\mathbb{Z}_p$...

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### subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose mod-2 image is the symmetric group

Let $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ be a closed subgroup of the symplectic group over the $2$-adic integers whose image under the mod-$2$ homomorphism $\pi : \mathrm{Sp}_{2g}(\mathbb{Z}_2)...

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58 views

### Representations theory of Groups with compact quotient

Let $G$ be a locally profinite group and $H$ a closed normal subgroup of $G$, with $G/H$ a profinite group.
If $\rho$ is an irreducible (smooth) representation of $G$ what can we say about the ...

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189 views

### On groups with finite pro-$p$ completion for all primes $p$

Say that a group has Property X if its pro-$p$-completion is finite for every prime $p$. For instance, every perfect group has Property X.
Is there a finitely generated, residually finite group $G$ ...

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268 views

### When does p-profinite completion commutes with maps from a $p$-finite space?

background
Let $\mathcal{S}$ be the ($\infty$-)category of spaces and $\mathcal{S}_{p-\text{finite}}$ the full subcategory spanned by the $p$-finite spaces (that is, the spaces with finitely many ...

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### Every profinite group is a quotient of a profinite free group by a normal subgroup that is free profinite?

It is well known that any group is a quotient a free group by a normal subgroup that is free. More precisely if $G$ is a group the exists a short exact sequence of groups
$$1\rightarrow F^{'}\...

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### Profinite extension of a Lie group

Let $H,G,K$ be three topological groups, we say that $G$ is an extension of $K$ by $H$ if the following short sequence
$$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0$$
is exact. (If $H$ is a ...

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### Are there open subgroups of $SL_2(\widehat{\mathbb{Z}})$ which are $GL_2(\widehat{\mathbb{Z}})$-conjugate, but not $SL_2$-conjugate?

I apologize if this is too obvious, but I figure it must have a quick answer.
Are there open subgroups $\Gamma\le SL_2(\widehat{\mathbb{Z}})$ which are conjugate in $GL_2(\widehat{\mathbb{Z}})$, but ...

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128 views

### Problem with a proof of Wilson's 'Profinite groups'

(Crossposted on StackExchange Mathematics: https://math.stackexchange.com/questions/2391626/problem-with-a-proof-of-wilsons-profinite-groups)
I need help with the proof of Proposition (3.1.3) given ...

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353 views

### Does $GL_2(\widehat{\mathbb{Z}})$ contain a dense finitely generated subgroup?

It's well known that $SL_2(\widehat{\mathbb{Z}})$ contains $SL_2(\mathbb{Z})$ as a dense and finitely generated subgroup. However, $GL_2(\mathbb{Z})$ is not dense in $GL_2(\widehat{\mathbb{Z}})$, ...

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### What is the probability of generating a given procyclic subgroup in $\mathrm{Gal}(\bar{K}/K)$?

This question began as Why are procyclic subgroups of Galois groups of number fields free profinite?, which fizzled out, but which garnered some helpful comments from YCor.
Let $K$ be a field, take $\...

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### Why are procyclic subgroups of Galois groups of number fields free profinite?

On p832 of Coombes, Harbater - Hurwitz familes and arithmetic Galois groups, the following is claimed:
Let $K$ be a number field, take $1 \neq \omega \in \mathrm{Gal}(\bar{\mathbb{Q}}/K)$, and let $...

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### Inverse limits and first isomorphism theorem for compact topological groups

This question was originally asked on MathSE here.
I have a problem with Proposition (1.2.1) from J. Wilson's book 'Profinite Groups'
The proposition is the following:
Let $(G, \varphi_i : G \to ...

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329 views

### Subgroups and quotients of an abelian pro-finite group

It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$.
I'm wondering whether there is a counterpart for profinite groups.
For example is it true ...

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252 views

### Does each compact topological group admit a discontinuous homomorphism to a Polish group?

A compact topological group $G$ is called Van der Waerden if each homomorphism $h:G\to K$ to a compact topological group is continuous. By a classical result of Van der Waerden (1933) the groups $SO(...

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### Is there a residually finite non-elementary hyperbolic group whose profinite completion is boundedly generated?

Is there a residually finite hyperbolic group $G$ that is not virtually cyclic, such that there exists finitely many procyclic closed subgroups $C_1, \dots, C_n$ of the profinite completion $\hat{G}$ ...