Questions tagged [profinite-groups]
For questions regarding profinite groups and their properties.
217
questions
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Relation between subgroup topology and Krull topology for an intermediate field of a Galois extension
Let $E/K$ be a Galois extension and let $F$ be an intermediate field such that $K\subseteq F\subseteq E$. Then $E/F$ is a Galois extension too and $H=\mbox{Gal}(E/F)$ is a closed subgroup of $G=\mbox{...
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1answer
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Automorphism group of rooted tree is a profinite group
I'm working with the book Self-similar Groups by Volodymyr Nekrashevych. In the chapter $1$ he statements the following proposition:
We have an equality $St_{Aut(X^*)}(n) = RiSt_{Aut(X^*)}(n)$. The ...
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7 views
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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1answer
49 views
Convergence in $\hat{\mathbb{Z}}$
The following is part of an exercise from Lenstra's Galois theory for schemes..
Let $a=\frac{b}{c}\in \mathbf{Q}^\times$, $n\in \widehat{\mathbf{Z}}^\times$.
Prove that there exists a sequence of ...
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1answer
30 views
Is the system of subgroups whose quotient is finitely generated, an inverse system?
In a similar fashion to the construction of the profinite completion, let $G$ be a group and let $$\mathfrak{M}=\{H\trianglelefteq G\;|\;G/H\text{ is finitely generated}\}$$
ordered by reverse ...
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1answer
42 views
Reference for properties of absolute Galois group of local field
Let $K$ be a local field. Let $K^{nr}$ and $K^t$ be its maximal unramified and tamely ramified extensions, respectively. One can show that $\operatorname{Gal}(K^{nr}/K) \cong \widehat{\mathbb Z}$ and ...
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1answer
28 views
In profinite group $G$ with inverse system $(G_\alpha)%$, how the kernel of projection map becomes the basis of neighborhood for $1$ in $G$?
In profinite group $G$ with inverse system $(G_\alpha,\phi_{\alpha,\beta} )%$, how the kernel of projection map $G\rightarrow G_\alpha$becomes the basis of neighborhood for $1$ in $G$?
I couldn't ...
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80 views
Galois group of infinite extension generated by all square roots of rationals
Let $L$ be the field obtained by adjoining all the square roots of all rationals. We know that $L/\mathbf Q$ is an infinite Galois extension.
Prove that the map
\begin{align*}
f:G=\...
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1answer
22 views
Absolute Galois group is topologically generated by Frobenius map clarification
I came across the following while reading a text on elliptic curves:
For the $q$th power Frobenius morphism on an elliptic curve $E$,
$\phi: E \rightarrow E$ given by $\phi(x,y) = (x^q, y^q)$,the ...
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25 views
$\text{Aut}(F_b^{fin})=\widetilde{\pi_1(X,b)}$
Given a space $X$ and fix a point $b\in X$, consider the functor $F_b^{fin}: \text{FinCov}(X)\to\text{FinSet}$ sending finite covering spaces to their (finite) fibers above $b$. Why is $\text{Aut}(F_b^...
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1answer
36 views
nilpotent uniform pro-$p$ groups of dimension 2
I read in a paper that all nilpotent uniform pro-$p$ groups of dimension $\leq 2$ are Abelian (Prime Decomposition and the Iwasawa $\mu$-Invariant by Hajir-Maire, Math. Proc. of the Camb. Phil. Soc. (...
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25 views
Finite-codimension ideal in profinite Lie algebras
Given a profinite group, it is well-known that any finite-index subgroup contains a finite-index normal subgroup, namely its core.
Does anything similar hold for profinite Lie algebras? More ...
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35 views
Equivalence of categories, related to Galois descent
I'm trying to understand an example (A.64) in the appendix of Milne's 2017 book on algebraic groups. It goes like this.
Let $K/k$ be a Galois extension with Galois group $\Gamma$. We want to show ...
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10 views
Profinite free modules and intersections
This is the part (c) in Exercise 5.2.4 of Luis Ribes' and Pavel Zalesskii's book Profinite Groups.
Let $\Lambda$ be a profinite ring, $(X, *)$ be a pointed profinite
space and $(Y_i,*)$ be a ...
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15 views
Example of finitely generated profinite group
Could someone please give an example of a finitely generated profinite group G such that its Frattini subgroup is not open? or a finitely generated profinite group that it is not an pro-p group?
...
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1answer
22 views
Factorization in a non-surjective inverse system of (pro)-finite spaces
Let $I$ be any directed poset and $\{X_i,\phi_{ij},I\}$ be an inverse system of profinite spaces with continuous maps $\phi_{ij}\colon X_i \to X_j$ whenever $i\geq j$ in $I$. Let $$X = \varprojlim_{i \...
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1answer
42 views
Compact p-adic analytic groups
It is a classical fact that a topological group $G$ admits the structure of $p$-adic analytic group iff it contains an open subgroup which is pro-p uniformly powerful. I was reading the related ...
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30 views
Strict continuous homomorphism between $\mathbb{Z}_p$-modules
I am reading the paper "Cohomology of p-adics analytic groups" of P. Symonds and T. Weygel and I have a question about a definition.
In section $2$ they introduce 3 categories of modules:
$\...
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0answers
33 views
Neighbourhood base consisting of open normal subgroups
I am working on profinite limits and the definition in Neukirch, Algebraic Number Theory using that a profinite limit is a topological group $G$ that is Hausdorff, compact and which has a ...
5
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1answer
97 views
About the definition of powerful p-groups
I am reading "Analytic pro-$p$ groups" by Dixon, Du Sautoy, Mann and Segal.
They define $G$ a finite $p$-group to be powerful if $[G,G]\leq G^p$ for $p$ odd but in the case $p=2$ they require $[G,G]\...
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1answer
29 views
Example of a surjective morphism of representations that is not surjective on the induced smooth representations
Let $G$ be a locally profinite group. Given an (abstract) representation $V$ of $G$, that is a $\mathbb C$-vector space $V$ together with a group homomorphism $G\rightarrow \operatorname{Aut}_{\mathbb ...
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1answer
32 views
Does $\mathbb Z/p\mathbb Z$ a free abelian pro-p group?
As far as I know, $\mathbb Z/p\mathbb Z$ ($p$ is prime) is cyclic and so it's abelian.
It is obviously a p-group, hence it is pro-p.
And it is free, for its generator, $\langle1\rangle$, has no ...
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27 views
Mackey theory in the setting of locally profinite groups
Is there an analogue of [1, Thm. 10.23] for ordinary smooth representations of a locally profinite group (l.p.g. for short)?
I have already checked introductory chapters of some classical references ...
3
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1answer
74 views
Pontryagin Duality [closed]
A pro-p group is a profinite group in which every open normal subgroup has index equal to some power of p. Suppose G is pro-p Group. Then, Whether there is a natural bijection between $Hom_{cont}(G, \...
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67 views
Ergodicity on the Profinite Integers versus Ergodicity on the $p$-adic integers
Let $\mathbb{P}$ denote the set of prime numbers, let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and let: $$\widetilde{\mathbb{Z}}\overset{\textrm{def}}{=}\prod_{p\in\mathbb{P}}\mathbb{Z}_{...
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0answers
80 views
Question about Neukirch's book Cohomology of Number Fields
Let $G$ be a profinite group. $A$ and $B$ are discrete G-modules. If $A^U=A$ for some open subgroup $U \subseteq G,$ then why Hom(A,B) is a discrete G-module. $\big( g(\phi)(a) = g(\phi(g^{-1}(a))) \...
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42 views
Isomorphism between Topological Groups.
Let $G$ is an abelian profinite group and $G=\varprojlim G_i$ (all $G_i$ are finite).
Then why $\hom(\varprojlim G_i,\mathbb{Q}/\mathbb{Z}) = \varinjlim\hom(G_i,\mathbb{Q}/\mathbb{Z})$ as ...
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Definition of Profinite Groups.
We know that totally disconnected topological groups are Hausdorff. Then why we are considering both Hausdorff and totally disconnected conditions in the definition of profinite groups.
Topological ...
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2answers
85 views
$p^i\mathbb {Z}_p$ is an open subgroup of $\mathbb {Z}_p$ of index $p^i$, where $\mathbb {Z}_p$ is the group of p-adic integers
I can't seem to understand the logic behind the following quote:
For each $i$, $p^i\mathbb {Z}_p$ is the kernel of the projection $\pi_i:\mathbb {Z}_p\rightarrow\mathbb Z/p^i\mathbb Z$.
Thus, $p^...
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56 views
continuous group homomorphism profinite/discrete topology
If $f:G\to H$ is a homomorphism of groups, $G$ is profinite, and $H$ is an infinite group with discrete topology is $f$ always continuous? If this is too difficult: What if $G=\mathbb Z_p$ is the ring ...
3
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1answer
95 views
When is the profinite completion of a centerless group itself centerless?
Having a centerless profinite completion leads to some nice properties. For example, given a short exact sequence
$$1\to A\to B\to C\to 1$$
where $A$ is finitely generated and $\hat{A}$ has trivial ...
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1answer
25 views
Show that the $\Gamma_1$ of $\mathrm{SL}_n(\mathbb{Z}_p)$ is pro-$p$ and finitely generated.
Problem. Fix a prime $p$ and a positive integer $n$. For each $j$ put
$$\Gamma_{j} = \{g \in \mathrm{SL}_{n}(\mathbb{Z}_{p}) \mid g \equiv 1_{n} \text{ mod }p^{j}\}.$$
(i) Show that $\Gamma_{1}$...
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0answers
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$\varprojlim(SL_n(Z)/K_n(p^i))_{i \in \mathbb{N}} \simeq SL_n(Z_p)$ and $\varprojlim(SL_n(Z)/K_n(m))_{m \in \mathbb{N}} \simeq SL_n(\hat{Z})$
Problem. Show that the natural map $\mathrm{SL}_{n}(\mathbb{Z}) \to \mathrm{SL}_{n}(\mathbb{Z}/m\mathbb{Z})$ is surjective, for all $m$ and $n$. Denoting its kernel by $K_{n}(m)$, show that
$$\...
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1answer
196 views
limits of powers of a matrix with entries in p-adic numbers; galois representation of finite fields
Let $u$ be a $d \times d$ matrix with coefficients in $\mathbb{Q}_\ell$, the $\ell$-adic numbers. Let $n \in \widehat{\mathbb{Z}}$. Something I'm reading claims (and says it's easy and leaves the ...
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1answer
97 views
Open normal subgroups of a profinite group form a basis for the open neighborhoods of 1
In the following link:
An equivalent definition of the profinite group
I'm having trouble understanding the following quote from the answer:
So if you replace each subgroup in a basis by the ...
2
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1answer
29 views
Two versions of Baire Category Theorem for Profinite Groups
Let $G$ be a profinite group:
Theorem 1. Let $C_{1},C_{2},...$ be a countably infinite set of nonempty closed subsets of $G$ having empty interior. Then
$$G \neq \bigcup_{n}^{\infty}C_{i}.$$
...
3
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2answers
68 views
Subset of a subgroup is not closed under group actions
In the following image (from "Field Arithmetic by Fried & Jarden" Page 6, Lemma 1.2.2(b)), red rectangle,
I'm trying to figure out why it's right to claim $h^{-1} \in H$.
I thought the ...
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1answer
51 views
A subset $C$ of a pro-finite group is closed iff $C$ is compact
I'm trying to prove both directions..
I think I managed to prove one: if $C$ is compact then $C$ is closed.
Feel free to correct me:
Every pro-finite group is compact, Hausdorff, and has a basis ...
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2answers
53 views
Galois group of $\overline{F}/F$
Let $K$ be the algebraic closure of $F$ where $F$ is a finite field. Show that $Gal(K/F) \simeq \hat{\mathbb{Z}}$.
I know that $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}_{n}$, so its enough to show ...
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23 views
Isomorphism between $\mathrm{Gal}(L_{1}L_{2}/L_{2}) \to \mathrm{Gal}(L_{1}/L_{1} \cap L_{2})$ (infinite extensions).
Let $K/F$ be a field extension. If $L_{1}$ and $L_{2}$ are between fields, prove that if $L_{1}/F$ is Galois, then $L_{1}L_{2}/L_{2}$ is Galois and there is a natural isomorphism
$$\mathrm{Gal}(L_{1}...
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1answer
655 views
Krull Topology on Galois Groups (two equivalent(?) definitions).
I'm starting to study Infinite Galois Theory and its relation with Profinite Groups, but I'm having troubles with basic definitions.
Definition 1. Let $K/F$ a Galois extension. Write
$$\mathcal{F}...
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1answer
259 views
The multiplicative group of units in $\mathbb{Z}_{p}$ is isomorphic to $\mathbb{Z}_{p} \times C_{n}$ with $n=\max\{p-1,2\}$.
Problem.
(a) Write down explicitly the rules for addition and multiplication in $\mathbb{Z}_{p}$.
(b) Show that $(\mathbb{Z}_{p^{i}},\varphi_{ij})$ where $\varphi:\mathbb{Z}_{p^{j}} \to \mathbb{...
1
vote
1answer
58 views
Elements conjugate in profinite completion
Problem. Let $G$ be a residually finite group, and identify $G$ with its image under the canonical map to its profinite completion $\hat{G}$. Let $x,y \in G$. Prove that the following conditions are ...
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1answer
46 views
If $H$ is a closed subgroup of a profinte group $G$, then $H$ is the inverse limit of the open subgroups of $G$ containing $H$.
Show that if $H$ is a closed subgroup of a profinte group $G$, then $H$ is the inverse limit of the open subgroups of $G$ containing $H$.
I was able to show that $H$ is profinite. I use a ...
5
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1answer
84 views
Book-recommendations “Profinite Groups”
I started studying Profinite Groups a few weeks ago. I'm using the book "Profinite Groups" by Wilson as a basis, but the book is not clear enough sometimes (probably because I'm studying it for the ...
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0answers
34 views
How is the base of the topology on a profinite group?
If $G$ and $H$ are profinite groups and $\varphi: G \to H$ is a homomorphism, then $\varphi$ is continuous iff the inverse image of any open normal subgroup in $H$ is open in $G$.
If $\varphi$ is ...
3
votes
1answer
60 views
If $(X_{i}, \varphi_{ij})$ is a inverse system of nonempty sets and surjective maps, then the inverse limit is nonempty (proof verification)
I had a troubles with this
problem, so I thought it would be important to write my solution because I'm not sure of all the details.
Since I already accepted the answer, I can not open a bounty to ...
0
votes
1answer
75 views
If $(X_{i}, \varphi_{ij})$ is a inverse system of nonempty sets and surjective maps, then the inverse limit is nonempty.
Problem. Prove that if $(X_{i}, \varphi_{ij})$ is a inverse system of nonempty sets and surjective maps indexed by a countable direct set, then the inverse limit is nonempty.
I proved some similar ...
5
votes
1answer
119 views
Does the Burnside $\mathbb Q$-algebra $A$ of a group depend only on $\dim_{\mathbb Q}A$?
The Burnside $\mathbb Q$-algebra $\mathbb QB(G)$ of a group $G$ is usually considered only when $G$ is finite; see Section 3.1 of the text
[1] Serge Bouc, https://pdfs.semanticscholar.org/aff3/...
4
votes
1answer
53 views
finiteness of an abelian topological group
Let A be an abelian (Hausdorff) topological group. Assume that
(1) the set of its torsion elements, and
(2) a finitely generated subgroup
are dense subsets of A.
My question: must A be finite?
(...
