Questions tagged [profinite-groups]

For questions regarding profinite groups and their properties.

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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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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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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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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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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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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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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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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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$\text{Aut}(F_b^{fin})=\widetilde{\pi_1(X,b)}$

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 $\text{Aut}(F_b^...
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nilpotent uniform pro-$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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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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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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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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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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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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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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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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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 ...
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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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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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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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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 ...
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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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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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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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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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$p^i\mathbb {Z}_p$ 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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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 ...
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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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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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$\varprojlim(SL_n(Z)/K_n(p^i))_{i \in \mathbb{N}} \simeq SL_n(Z_p)$ 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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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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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 ...
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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 $$G \neq \bigcup_{n}^{\infty}C_{i}.$$ ...
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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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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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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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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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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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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{...
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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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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 ...
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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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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 ...
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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 ...
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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 ...
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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/...
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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? (...

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