Questions tagged [profinite-groups]
The profinite-groups tag has no usage guidance.
323 questions
4
votes
0
answers
252
views
+300
Is a pro-algebraic group over $\mathbb{Q}_p$ with Galois action the inverse limit of Galois-equivariant quotients?
Let $\mathcal{G}$ be a pro-algebraic group over $\mathbb{Q}_p$ with a continuous action of $G_K$ for a field $K$ (if $\mathcal{G}$ were an abelian unipotent group, this is precisely a $p$-adic Galois ...
- 15.4k
0
votes
1
answer
64
views
Transitive map on a profinite group
Let $f$ be a continuous endomorphism of a compact Hausdorff totally disconnected topological group $G$ and let $H$ be a closed normal subgroup of G such that $f(H)\subseteq H$ and with $\mu(H)=0$ ...
- 361
4
votes
0
answers
116
views
Are there Mackey decompositions for closed subgroups of profinite groups?
$\DeclareMathOperator\ind{ind}\DeclareMathOperator\res{res}$
My question: what is known or expected to be true about the existence of a Mackey decomposition formula in the case of closed subgroups of ...
- 1,335
3
votes
0
answers
89
views
Which elements in $\mathrm{Aut}(\widehat{F_2})$ preserve the procyclic subgroup generated by the commutator $c=[a,b]$?
Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
- 605
4
votes
0
answers
97
views
Characterization of Vilenkin group
It is shown in [1, Section 1] by C.W. Onneweer that every infinite compact, metrizable, zero-dimensional commutative group is a Vilenkin group. My question is does this implication also hold if we ...
- 85
9
votes
1
answer
223
views
$p$-adic analytic pro-$p$ group satisfies a pro-$p$ identity?
Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity
$w$ if for every homomorphism $ f:...
- 667
3
votes
1
answer
133
views
Any Sylow pro-$p$ subgroup of a topologically finitely generated profinite group is also topologically finitely generated?
It's proved in Oltikar and Ribes - On prosupersolvable groups that any Sylow pro-$p$ subgroup of a topologically finitely generated prosupersolvable group is also topologically finitely generated. It ...
- 667
5
votes
1
answer
156
views
The inverse limit of a sequence of ring surjections commutes with taking difference subsets of the respective units & gluing in some primes?
Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings.
Then if $f_{n,n+1}: \Bbb{Z}/p_{...
- 260
3
votes
1
answer
159
views
Does the fundamental group of a compact 3-manifold induce full profinite topology on the fundamental group of its boundary?
Let $M^3$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $S\subseteq\partial M$ be one of its boundary components. Does $\pi_1(M)$ induce the full profinite ...
- 605
3
votes
1
answer
423
views
Groups with no homomorphisms onto $\mathbb{Z}/p\mathbb{Z}$
Does there exist any term for finite groups with no non-trivial homomorphisms into $\mathbb{Z}/p\mathbb{Z}$ for a fixed prime $p$, or any term related to this property (so that I could write "...
- 16.9k
2
votes
0
answers
115
views
Test words in free profinite groups
Let $G$ be a group. An element $g \in G$ is said to be a test element if any endomorphism $\phi$ of $G$ such that $\phi(g) = g$ is an automorphism. The free group $F_2$ of rank $2$ is generated by $...
- 355
3
votes
1
answer
222
views
A question about coprime automorphisms of profinite groups
Let $p$ a prime. A finite group is a $p'$-group if its order is prime to $p$. Let $A$ be a finite $p'$-group of automorphisms of a finite $p$-group $G$. Suppose that $A$ is a non-cyclic abelian group. ...
- 863
1
vote
0
answers
96
views
Definition of free profinite product of infinitely many groups
If we have profinite groups $G_1,...,G_n$ we can define its free profinite product in the natural way. But this natural definition (similar to the abstract case but in the category of profinite groups)...
- 329
1
vote
0
answers
104
views
Infinite closed subgroup of ${\rm SL}_{n}(\mathbb{F}_{p}[[T]])$ with full residual image
Let $\mathbb{F}_{p}$ be a finite field of order $p$, $\mathbb{Z}_p$ be the ring of $p$-adic integers and $\mathbb{F}_{p}[[T]]$ be the ring of formal power series over $\mathbb{F}_{p}$. For $p\geq 5$, ...
- 667
2
votes
0
answers
71
views
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}...
- 667
4
votes
1
answer
582
views
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 ...
- 460
3
votes
0
answers
383
views
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 ...
- 667
4
votes
0
answers
110
views
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 ...
- 21.8k
1
vote
0
answers
61
views
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, ...
user30211
1
vote
0
answers
63
views
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 ...
- 11
3
votes
0
answers
69
views
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 $\mathbb{F}_p$ ...
- 31
1
vote
0
answers
126
views
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-$p$ ...
- 667
2
votes
0
answers
106
views
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 ...
- 667
1
vote
0
answers
126
views
$p'$-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 $G$. ...
- 667
1
vote
0
answers
84
views
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 ...
- 667
2
votes
0
answers
165
views
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}...
- 667
2
votes
1
answer
96
views
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 $G$...
- 329
3
votes
0
answers
163
views
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 ...
- 253
8
votes
1
answer
181
views
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) ...
- 253
6
votes
2
answers
188
views
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 $\...
- 2,519
4
votes
1
answer
233
views
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 ...
- 253
4
votes
0
answers
181
views
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 ...
- 863
1
vote
0
answers
98
views
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 $. In ...
- 863
0
votes
0
answers
54
views
Existence of maximal topologically characteristic subgroup of infinite index of pro-$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 ...
- 667
7
votes
1
answer
347
views
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
1
answer
150
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
89
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
173
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 ...
- 667
1
vote
0
answers
120
views
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 $\chi(H)$, $\chi(G)$,...
- 1,013
2
votes
0
answers
59
views
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 $f: X \to F$ ...
- 329
2
votes
0
answers
171
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\...
- 667
3
votes
1
answer
187
views
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 $B(2,3)$. ...
- 329
3
votes
0
answers
143
views
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 ...
- 863
2
votes
1
answer
336
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)...
- 329
5
votes
1
answer
496
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 ...
- 531
3
votes
0
answers
242
views
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 ...
- 667
1
vote
1
answer
95
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 ...
- 863
2
votes
1
answer
238
views
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?
$$...
- 769
2
votes
1
answer
203
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 $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p\mathbb{...
- 863
8
votes
0
answers
296
views
Quantizing the size of a pro-$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 ...
- 685