# Questions tagged [profinite-groups]

The profinite-groups tag has no usage guidance.

310
questions

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### On the conjugation action on the first congruence subgroup of special linear group over $p$-adic fields

Let $\mathcal{O}_{L}$ be the ring of integers of a finite extension $L$ of $p$-adic number fields $\mathbb{Q}_{p}$ where $p$ is an odd prime. Let $\mathfrak{m}_{L}$ be the maximal ideal of $\mathcal{O}...

4
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1
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501
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### Subgroup of p-adic units

Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$
be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$.
We give it the product ...

3
votes

0
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322
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### Semidirect product in inverse Galois problem

Let $L/\mathbb{Q}$ (resp. $K/\mathbb{Q}$) be a Galois extension of rational number field $\mathbb{Q}$ with Galois group $P$ (resp. $H$) where $P$ is a second countable pro-$p$ group and $H$ is a ...

4
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### Hecke algebra $\mathcal{H}(K_1\backslash \mathrm{GL}_n(\mathbb{F})/K_1)$

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...

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### Matroid for Laurent series

I am trying to find a matroid for profinite rings which are the inverse limit of their finite quotients, and whose linearly independent elements are of the form $L((t_1,\dots,t_n))$.
To set this up, ...

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0
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49
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### Sylow subgroups of the free product of profinite groups

I am interested in the Sylow subgroups of the profinite completion of a free product of finite groups.
Is the following naive expectation true ? I assume things like this should be well-known, and am ...

3
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### Metrisable profinite groups

I do not understand on page 6 of Galois Cohomology from Serre, the comment after exercise 2) part d). He claims that taking G to be the dual of a countably dimensional vector space over $\mathbb{F}_p$ ...

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### Finitely generated torsion-free pro-$p$ subgroup of ${\rm GL}_{n}(\mathbb{F}_{p}[[T]])$ is solvable?

Let $\mathbb{F}_{p}$ be a finite field of order $p$, and $\mathbb{F}_{p}[[T]]$ be the ring of formal power series over $\mathbb{F}_{p}$. My question is the following:
Let $G$ be a closed pro-$p$ ...

2
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### Hereditarily just-infinite pro-$2$ groups

An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...

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122
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### $p'$-automorphisms of pro-$p$ groups

Let $p$ be a prime and $G$ be a finitely generated pro-$p$ group admitting a continuous automorphism $\phi$ of finite order relatively prime to $p$. Let $\Phi(G)$ denote the Frattini subgroup of $G$. ...

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### Non-Noetherian closed subgroups of ${\rm GL}_{n}(\mathbb{F}_{q}[[T]])$

Let $\mathbb{F}_{q}$ be a finite field of order $q$, and $\mathbb{F}_{q}[[T]]$ be the ring of formal power series over $\mathbb{F}_{q}$. We say that a profinite group $G$ is Noetherian if any closed ...

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146
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### Prime-to-$p$ quotients of ${\rm PSL}_{2}(\mathbb{Z}_{p})$

Let $p$ be a prime and $\mathbb{Z}_p$ the ring of $p$-adic integers. Let ${\rm PSL}_{2}(\mathbb{Z}_{p})={\rm SL}_{2}(\mathbb{Z}_{p})/\{\pm 1\}$ be the projective special linear group over $\mathbb{Z}...

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### If $F$ is a prosoluble subgroup of a free profinite product $\amalg G_i$ and $F \cap G_i^g$ is pro-$p$, is also $F$ pro-$p$?

There is a 1995 paper (Manusc. Math., DOI link) of Florian Pop where he proves the following:
Theorem. Let $G$ be the free product of profinite groups $G_i$ and $F$ a closed prosoluble subgroup of $G$...

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152
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### When is a group the same as its profinite completion

I'm working with the inner automorphism group of profinite quandles. A question I have yet to resolve is whether or not the inner automorphism group of a profinite quandle is necessarily profinite, or ...

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### Stone-topological/profinite equivalence for quandles

A quandle $(Q,\triangleleft,\triangleleft^{-1})$ is a set $Q$ with two binary operations $\triangleleft,\triangleleft^{-1}:Q\times Q\to Q$ such that the following hold for all $x,y,z\in Q$:
(Q1) ...

6
votes

2
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185
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### Agemo-of-agemo inclusions for p-groups

For a finite $p$-group $G$, let $\mho_i(G)$ denote the subgroup generated by $p^i$-powers of elements of $G$.
It is well-known that $\mho_i(\mho_j(G))$ can differ from $\mho_j(\mho_i(G))$ and from $\...

4
votes

1
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199
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### Profinite groups with isomorphic proper, dense subgroups are isomorphic

I am developing a sort of standard representation for profinite quandles. This involves profinite groups a lot, actually. In one part of my construction the filtered diagram used to construct a ...

4
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164
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### The order of the global Galois group

For any profinite group $G$, we can define the order of $G$ using the notion of supernatural number. Now let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group ...

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### Existence of countable dense normal subgroups of global Galois group

Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...

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49
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### Existence of maximal topologically characteristic subgroup of infinite index of pro-$p$ groups

Let $G$ be a topologically finitely generated infinite pro-$p$ group. Suppose that $G$ is not just-infinite. Does the group $G$ always have a maximal topologically characteristic subgroup of infinite ...

7
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### Topological generators for $\mathrm{SL}_2(\mathbf{Z}_p)$

$\DeclareMathOperator\SL{SL}$ Let $p>3$ and $G$ be an open subgroup of the special linear group $\SL_2(\mathbf{Z}_p)$ over the ring $\mathbf{Z}_p$ of $p$-adic integers. Suppose that $G$ is ...

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### Finite pro-$ p $ subgroups of $ {\rm SL}_{2}(\mathbb{F}[[T]]) $

Let $ p $ be an odd prime, $ \mathbb{F} $ a finite field of characterisitc $ p $ and $ \mathbb{F}[[T]] $ the formal power series over $ \mathbb{F} $. Let $ G $ be a pro-$ p $ subgroup of $ {\rm SL}_{2}...

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1
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147
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### Topological generators for the Sylow pro-$p$ subgroup of $\mathrm{SL}_2(\mathbf{Z}_p)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $G_2(\mathbf{Z}_p):=\begin{pmatrix}
1+p\mathbf{Z}_{p} & \mathbf{Z}_{p}\\
p\mathbf{Z}_{p} & 1+p \mathbf{Z}_{p}
\end{pmatrix}$. ...

1
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0
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86
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### A closed subgroup of $p$-adic analytic group having same dimension is open?

Let $G$ be $p$-adic analytic pro-$p$ group and $H$ a closed subgroup of $G$. Suppose that $G$ and $H$ have the same dimension as $p$-adic analytic groups.
Question: Is it true that $H$ is an open ...

3
votes

1
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169
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### Open conjugacy classes in a second countable profinite group

Let $G$ be a second countable profinite group, $g\in G$ and $g^G:=\{hgh^{-1}~|~h\in G\}$ the conjugacy class of $g$ in $G$. Theorem 3.2 in Wesolek's Conjugacy class conditions in locally compact ...

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116
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### Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups

While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...

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### Convergent condition in the definition of free profinite groups

I have a (perhaps simple) question about free pro-$C$ constructions.
Definition. Let $X$ be a set. The free pro-$C$ group over $X$ is a pro-$C$ group $F$ together a $1$-convergent map $f: X \to F$ ...

2
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169
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### When is an infinite pro-$p$ group generated by its torsions

Let $p$ be a prime and $\mathcal{O}=\mathbf{Z}_p$ or $\mathbf{F}_p[[T]]$, i.e. the ring of $p$-adic integers or the ring of formal power series over a finite field $\mathbf{F}_p$ of order $p$. Let $G\...

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162
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### Profinite completion of Baumslag-Solitar group as a profinite HNN-extension

I apologize if this question is basic in some sense. I was looking for an example of a non-proper HNN-extension and I found this.
In the comments, markvs mentioned the Baumslag-Solitar group $B(2,3)$. ...

3
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131
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### Just-infinite quotients of pro-$p$ groups that are linear over a complete Noetherian local ring

This question is a sequel to Quotients of pro-p
groups linear over a complete Noetherian local ring. Recall that an infinite pro-$p$ group is called just-infinite if it has no proper, infinite ...

2
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1
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258
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### Is the free profinite group (or pro-$p$) torsion-free?

Let $X$ be a set and $\widehat{F}(X)$ the restricted free profinite group on $X$. To get $\widehat{F}(X)$ we define a profinite topology on $F$ (the free abstract group on $X$) and take $\widehat{F}(X)...

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1
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### “Sheaf cohomology” of Galois groups

Apparently the Galois group $G$ of a Galois extension $E/F$ can be viewed as an “étale sheaf” on the set $X$ of intermediate Galois extensions equipped with an appropriate Grothendieck topology (see ...

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### Is insoluble $p$-adic analytic just-infinite pro-$p$ group torsion-free?

Recall that an infinite pro-$p$ group $G$ is called just-infinite if all non-trivial closed normal subgroup of $G$ have finite index.
Question: Let $G$ be an insoluble $p$-adic analytic just-infinite ...

1
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1
answer

89
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### Quotients of pro-$p$ groups linear over a complete Noetherian local ring

Let $R$ be complete Noetherian local ring with finite residue field $\mathbb{F}$ of characteristic $ p $. We say that a pro-$p$ group $G$ is linear over $R$ if it is isomorphic to a closed subgroup of ...

2
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1
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### For locally profinite groups $H\lhd G$, is there a spectral sequence $\newcommand\@[2]{{\rm Ext}_#1^{#2}(\pi_1,\pi_2)}H^p(G/H,\@Hq)\implies\@G{p+q}$?

Let $G$ be a locally profinite group and let $H$ be a closed normal subgroup. Let $\pi_1$ and $\pi_2$ be two smooth complex representations of $G$. Is there always a spectral sequence as follows?
$$...

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1
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### Finite quotients of $p$-adic congruence subgroups of $\operatorname{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime and let $ \SL^1_2(\mathbb{Z}_p)$ denote the kernel of the natrual surjective morphism $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p\mathbb{...

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### Quantizing the size of a pro-$p$ group

Let $p$ be a prime number and $G$ be a pro-$p$ group (not necessarily powerful). Let $\Omega$ denote the completed group algebra $\mathbb{F}_p[[G]]:=\varprojlim_N \mathbb{F}_p[G/N]$, where $N$ ranges ...

2
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### Structure theorem for finitely generated profinite abelian groups

Is there a structure theorem for finitely generated profinite abelian group like a structure theorem of f.g. abelian group?

2
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1
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132
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### The dimension of a torsion-free $p$-adic analytic group generated by two generators

$\DeclareMathOperator\GL{GL}$Let $G$ be a $2$-generator pro-$p$-group of finite rank, i.e. it is isomorphic to a closed subgroup of $\GL_d(\mathbb{Z}_p)$ for some integer $d$. Assume that $G$ is ...

5
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1
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### Classification of natural endomorphisms on finite groups

Any $z \in \widehat{\mathbb{Z}} = \lim_{n} \mathbb{Z}/n\mathbb{Z}$ defines an operation on all finite groups: if $G$ is a finite group and $g \in G$, say $g^n=1$, then map it to $g^{z_n}$. This ...

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### When a pro-$p$ group of finite rank can be embedded into the first congruence subgroup of ${\rm GL}_{N}(\mathbb{Z}_{p})$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ be a odd prime. We say that a pro-$p$ group has finite rank if it is isomorphic to a closed subgroup of $\GL_d(\mathbb{Z}_p)$ for some ...

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### Generators of of $p$-adic congruence subgroups of $\operatorname{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime, $i$ a fixed positive integer and let $\Gamma_i$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^i\mathbb{Z}_p)$. ...

2
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1
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### Nontrivial abelianization of torsion-free pro-$p$-group which contains a dense free subgroup is infinite?

Let $G$ be a topologically finitely generated pro-$p$-group. Assume that $G$ is torsion-free, it contains a dense free subgroup of infinite rank and the (topological) abelianization $G^{\text{ab}}$ of ...

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2
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157
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### Looking for an example of profinite groups

Is there a profinite group $G$ with a locally finite subgroup $H$ such that $\overline H$, the closure of $H$, is not torsion?

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### A question about a class of pro-$\mathcal X$-group

This question concerns the following lemma of this paper:
Lemma 2. Let $\mathcal X_1,\ldots,\mathcal X_n$ be classes of finite groups closed with respect to normal subgroups and subdirect
products ...

1
vote

1
answer

160
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### Openness of product of two open subgroups

Let $G$ be a profinite topological group with two closed subgroup $G_1$ and $G_2$. Suppose $G_1$ is normal in $G$ and $G=G_1G_2$. Let $H_i$ be an open subgroup in $G_i$ for $i=1,2$.
Question: Is $ ...

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### Endo reversible words

Let $w$ be a word in free group $F$ on finitely many generators. We will look at $w$ as word map on groups. It is clear that there exists an endomorphism $\phi$ of $F$ such that $\phi(w) = w^{-1}$ if ...

2
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1
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139
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### Subgroup growth of direct product

I have started reading about subgroup growth and, to my surprise, I haven't found a reference to whether direct products preserve subgroup growth.
Recall that, given a finitely generated group $G$, ...

5
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1
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662
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### Structure of a profinite group as a condensed set with an action of an open subgroup

Let $G$ be a profinite group and $H$ be an open subgroup. As a continuous $H$-topological space, we have $G=\coprod_{G/H} H$. Does this also hold as condensed sets, i.e. do we have an identification ...

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### An infinite profinite group such that any $\overline{\mathbb{F}_{p}((t))}$-adic representation has finite image

This question is a sequel to An infinite profinite group such that any $p$-adic representation has finite image
.
Fix a prime $ p $. We call an infinite profinite group $G$ a Boston group (with ...