Questions tagged [profinite-groups]
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289
questions
6
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249
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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 ...
0
votes
0
answers
68
views
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}...
0
votes
1
answer
131
views
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
vote
0
answers
69
views
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
answer
143
views
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 ...
- 289
1
vote
0
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109
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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 $\chi(H)$, $\chi(G)$,...
- 1,033
2
votes
0
answers
44
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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 $f: X \to F$ ...
- 267
2
votes
0
answers
145
views
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\...
- 289
3
votes
1
answer
101
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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 $B(2,3)$. ...
- 267
2
votes
0
answers
64
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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 ...
- 695
2
votes
1
answer
150
views
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)...
- 267
4
votes
1
answer
299
views
“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 ...
- 224
3
votes
0
answers
228
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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 ...
- 289
1
vote
1
answer
79
views
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 ...
- 695
2
votes
1
answer
208
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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?
$$...
- 465
2
votes
1
answer
176
views
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{...
- 695
8
votes
0
answers
262
views
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 ...
- 523
1
vote
0
answers
134
views
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?
- 599
2
votes
1
answer
105
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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 ...
- 289
5
votes
1
answer
397
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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 ...
- 58.8k
7
votes
1
answer
233
views
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 ...
- 289
1
vote
0
answers
119
views
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)$. ...
- 695
2
votes
1
answer
111
views
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 ...
- 289
1
vote
2
answers
150
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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?
- 2,108
0
votes
0
answers
73
views
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 ...
- 2,108
1
vote
1
answer
141
views
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 $ ...
- 367
7
votes
0
answers
106
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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 ...
- 213
2
votes
1
answer
115
views
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$, ...
- 541
5
votes
1
answer
569
views
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 ...
- 607
1
vote
0
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114
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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 ...
- 695
4
votes
2
answers
310
views
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 Fontaine-Mazur group (with respect to $ p $) if every continuous homomorphism $ G\to {\rm GL}_n(\overline{\mathbb{Q}}_p) $ has finite ...
- 695
3
votes
0
answers
245
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Non-abelian group cohomology, additional information
Let $G$ be a (profinite) group, and let $M$ be a non-abelian $G$-module.
We know how to construct reasonably $H^0(G,M)$ and $H^1(G,M)$ and it turns out that $H^1(G,M)$ is just a pointed set and not ...
- 361
5
votes
0
answers
139
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Can there be non-isomorphic fundamental groups of equivalent Galois categories?
It is known that if $(C, F)$ is a Galois category then there exists an equivalence $C \cong \pi_1(C, F)-FinSets$ between $C$ and the category of finite sets with continuous actions of $\pi_1(C, F) := ...
- 1,393
4
votes
0
answers
123
views
Proving that the fundamental group of a finite Galois category is profinite
This is sort of a cross-posting of this question of mine over on math.SE.
Suppose that I am given a finite Galois category $(\mathcal{G}, F)$, i.e. a Galois category in the sense of SGA 1. One of the ...
- 1,393
2
votes
1
answer
302
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Existence of regular semisimple elements in linear group over local field
Let $ L $ be a finite extension of $p$-adic numbers $ \mathbb{Q}_p $. Let $ \text{GL}_{n}(L) $ denote the general linear group $ \text{GL}_{n}(L) $ over $L$ equipped with the topology induced from the ...
- 695
2
votes
0
answers
128
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Artin map and profinite completion of the idèles
One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which ...
- 2,402
1
vote
0
answers
106
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Direct limit of groups rings of finite quotients of a profinite group
Background:
Let $G$ be a profinite group, for $M \leqslant N$ open normal subgroups we have the projection map $p_{M,N} \colon G/M \to G/N$
which induces a transfer map on rational group rings
$$
p_{M,...
- 371
2
votes
1
answer
236
views
Difference between definitions of continuous action, profinite case
My setting is the following : let $G$ be a topological group and $X$ be a topological space. I have the head filled with two possible definitions for a continuous action of $G$ on $X$.
The first could ...
0
votes
1
answer
102
views
Examples of non-proper profinite HNN extensions
We define a profinite HNN extension as the profinite completion of the abstract HNN extension. In the abstract case, the homomorphim of the base group to the HNN extension is always a monomorphism. ...
1
vote
0
answers
172
views
Fields such that every finite Galois extension is solvable
What are the fields such that every finite Galois extension is solvable?
We have algebraically closed fields, real closed fields, p-adic fields. Anything else?
A more pointed question after comments:
...
- 21
1
vote
1
answer
123
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Free partners in semi-direct products
Let $G = N \rtimes K$ be a semi-direct product of groups and suppose that $K$ is a finite group. Call the set $\mathcal{F} = \{ \alpha \in G \mid \langle \alpha, K \rangle = \langle \alpha \rangle \...
5
votes
0
answers
186
views
When is the profinite completion of a Noetherian group ring also Noetherian?
Let $G$ be a group, and let $\mathbb{Z}[G]$ denote its group ring. Its profinite completion is the inverse limit over all ideals of finite index. By Benjamin Steinberg's answer here, this profinite ...
- 6,253
4
votes
2
answers
278
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Topology on the hom space between profinite groups
$\DeclareMathOperator\Hom{Hom}$Let $G,H$ be profinite groups. Let $\Hom(G,H)$ be the set of continuous group homomorphisms, equipped with the compact-open topology. I'd like to understand the ...
- 6,253
6
votes
1
answer
363
views
Irreducible representations of product of profinite groups
It is a standard fact in the representation theory of finite groups that for $G,H$ finite groups, all of the irreducible representations of $G \times H$ are the external tensor product of irreps of $G$...
- 692
4
votes
0
answers
139
views
Confusion with self-dual representations of $\mathrm{GL}_n$ over a $p$-adic field
The following surely is kind of a trivial question, but it keeps me confused. It concerns a detail in Lust and Stevens' paper "On depth zero L-packets for classical groups" London Math. Soc. ...
- 465
3
votes
0
answers
88
views
Projective limit of copies of same group w.r.t. some fixed endomorphism
In our study of automorphism groups of transcendental field extensions, we have encountered the situation where we have a group $F$ together with an endomorphism $\alpha \colon F \to F$, resulting in ...
- 5,654
2
votes
0
answers
78
views
Normal subgroups of prosupersolvable groups
Let $G$ be a finite supersolvable group, and if $p$ is the biggest prime dividing $\vert G \vert$. Then $G$ has normal subgroups of order every possible power of $p$.
Analogous statement in case of ...
- 213
2
votes
0
answers
98
views
Profinite projective and free modules
I am studying cohomology of profinite groups and the following question came to my mind: suppose we have $G$ a pro-$p$ group which is Poincaré Dual of dimension $d$. This means that $\mathbb{Z}_p$ as ...
- 717
6
votes
1
answer
210
views
Top cohomology of profinite Poincaré duality group
The paper "Cohomology of p-adic analytic groups" by Symonds and Weigel is considered one of the main references for continuous cohomology of profinite groups. There is a passage I do not ...
- 717
3
votes
1
answer
205
views
Absolute Galois group with unique closed non-open subgroup
Is there an absolute Galois group that is not a subgroup of $\hat{\mathbb{Z}}$ and that has one and only one closed non-open subgroup?
- 53