Questions tagged [pro-p-groups]

For questions concerning pro-$p$ groups. These groups arise naturally in topology, algebraic number theory or Galois theory and are a special case of pro-finite groups.

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79 views

Pro-finite completion of p-adic Lie groups

Consider a $p$-adic Lie group $G$. My question is if the pro-finite completion $\hat{G}$ is a $p$-adic Lie group. First we note that since $$\hat{G}=\text{lim}_{N\subset G} G/N$$ where the limit ...
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1answer
38 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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110 views

Finite intersection property for sets containing generating elements of derived subgroups of quotients

What I need to prove is a consequence of the following theorem. Theorem A. Let $G$ be a finite $p$-group and suppose that its derived subgroup $G'$ is generated by 2 elements. Then there exists $x\...
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1answer
44 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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19 views

A homomorphism from a pro-$p$ group to $\Bbb Z/p\Bbb Z$.

I met this question while reading Serre's book Galois cohomology. In order to prove a property that the $p$-cohomological dimension of a profinite group $G$ is zero if and only if the order of $G$ is ...
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
33 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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2answers
64 views

Normal closure of powerfully embedded subgroups.

I'm reading a paper of Lubotzky and Mann (J. Algebra 105, 1987, 484-505), and Im doubtful in a proof. Proposition. Let $N$ powerfully embedded subgroup of $G$. If $N$ is the normal closure of some ...
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85 views

Dimension of pro-$p$ group

Problem. Let $G$ be a pro-$p$ group of finite rank. Prove that $$\mathrm{dim}(G) = \lim_{k \to \infty}\frac{\log_{p}|G:G^{p^k}|}{k}.$$ By definition $$\mathrm{dim}(G) = \mathrm{d}(H)$$ where $H$ is ...
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94 views

Uniform powerful pro-$p$ group isomorphic to $Z_{p}^{d}$

Problem. Let $G$ be a uniform pro-$p$ group and suppose that $G$ has an abelian open normal subgroup. Show that $G/Z(G)$ is finite, deduce that in fact $G \simeq \mathbb{Z}_{p}^{d}$ for some $d$. (...
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1answer
52 views

Meta-procyclic groups.

Theorem. A pro-$p$ group is meta-procyclic iff it is a inverse limit of metacyclic $p$-groups. Proof. Let $G$ be a meta-procyclic pro-$p$ group with normal subgroup $N$ and let $M$ be an open normal ...
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1answer
47 views

A powerful $2$-generated $p$-group is metacyclic

Problem. A powerful $2$-generated $p$-group is metacyclic. My book gives a hint: "prove that $G'$ is cyclic." Using this hint, I can solve the problem, but I'm in troubles to prove it. My attempt is ...
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138 views

Index of congruence subgroups in $GL_2(\mathbb{Z}_p)$ modulo their centers.

Let $\Gamma_i$ be the set of matrices in $GL_2(\mathbb{Z}_p)$ which are congruent to $1$ modulo $p^i$, that is they are the congruence subgroups. I know that $\Gamma_i$ is a pro-$p$ group and $\...
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1answer
114 views

Let G be $\mathbb Z_p\times\dots\times \mathbb Z_p$ . Find A(G).

Let G be $\underbrace{\mathbb Z_p\times\dots\times \mathbb Z_p}_{n \text{ times}}$. Find $A(G)$. I know that $A(G)\cong GL_n(\mathbb Z_p)$. I prove it by taking $\varphi$ from $A(G)$ and show that ...
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27 views

Extension of continuous map on group ring to a map on the complete group algebra

I'm reading the book "Galois Theory of $p$-Extensions" by Helmut Koch. And I can't understand the Theorem 7.2 of his book. The assumptions on the theorem is as follows : $G$ is a profinite group, $R$...
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57 views

Example Powerful Pro-$p$-Groups

I'm seeking for some nice examples for powerful pro-p-groups* for prime $p \neq 2$. By definition a powerful $p$- group $G$ is definined by following property: The commutator $[G,G]$ is contained ...
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1answer
171 views

What's the maximal pro 2 Galois extension unramified outside 2, 3 and infinity over Q?

I encounter a problem in my research: Let $L$ be the maximal pro-2 extension unramified outside $2, 3, \infty$ over $Q$, I hope I could know some information about the Galois group $Gal(L/Q)$. However,...
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1answer
72 views

Limit of quotients by $p^n$-th powers in $p$-adic fields

Let $K/\mathbb{Q}_p$ be a finite extension with normalized valuation $v_K$, let $\mathcal O_K$ be its ring of integers, and let $\mathfrak m_K$ be the maximal ideal of $\mathcal O_K$. Denote $U^N=1+\...
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1answer
133 views

Why is the first $p$-adic congruence subgroup a pro-$p$ group?

I am trying to see that $\Gamma_2$, defined as the kernel of the natural surjective map $\text{GL}_2(\mathbb Z_p)\to \text{GL}(\mathbb F_p)$ is a pro-$p$ group. So I'm trying to show that every ...
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2answers
366 views

Why is the group of principal units of a local field uniquely divisible by $n$?

I am reading a proof with the followings setup and claim. $K/F$ is a Galois extension of local fields with group $G$ of order $n = q^s$, where $q$ is prime and $s \geq 1$. Assume the maximal ideal $\...
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47 views

If the subgroup $H$ of $G$ is open in pro-$p$ topology, does it inherit the pro-$p$ topology?

Fix a prime $p$. Let $G$ be a group endowed with the pro-$p$ topology, and let $H$ be an open subgroup of $G$. I am trying to prove that the induced topology on $H$ is the pro-$p$ topology of $H$. ...
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2answers
466 views

Do there exist pro-$p$ groups with finite quotients of non $p$ power order?

We define a pro-$p$ group to be a projective (i.e. inverse) limit of $p$-groups. My question is exactly as stated in the title: If a subgroup $H$ of a pro-$p$ group $G$ has finite quotient, must $|...
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56 views

Proof that a particular subgroup is proper

I've been stuck on this for a long time ... I'm reading a textbook which simply states "this subgroup is proper" but it doesn't make sense to me. Context: I have a pro-$p$ group $G$, which just means ...
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1answer
479 views

Nontrivial examples of pro-$p$ groups

I only know a few examples of pro-$p$ groups. Of course the $p$-adics $\mathbf{Z}_p$, and any finite $p$-group. Congruence subgroups of $\text{GL}_n(\mathbf{Z}_p)$: e.g. $\Gamma_1:=\{g\in \text{SL}_n(...
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2answers
80 views

Show finite group is $p$-group given some structure of group

Let $G$ be a finite group. If there exists an $a\in G$ not equal to the identity such that for all $x\in G$,$\phi(x) = axa^{-1}=x^{p+1} $ is an automorphism of $G$ then $G$ is a $p$-group. This is ...
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1answer
447 views

Embed local Galois groups in global Galois group

Let $k$ be a global field, $p$ be a rational prime and let $S$ be a set of primes of $k$ with density $\delta(S) = 1$. Let $\mathfrak{p} \in S$ be a prime and denote by $k_\mathfrak{p}$ the completion ...
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1answer
345 views

Cohomological ($p$-)dimension of a pro-$p$ group

I have a question concerning the cohomological dimension and $p$-dimension of a pro-$p$-group. Let's first recall the definitions of that The cohomological dimension $cd \ G$ of a pro-finite group $G$...
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1answer
319 views

$p$-adic analytic group are closed subgroups of $GL_n(\mathbb{Z}_p)$ for some $n$

The article on pro-$p$-groups on Wikipedia tells us, that any $p$-adic analytic group can be found as a closed subgroup of $GL_n(\mathbb{Z}_p)$ for some $n \geq 0$. Do you have a reference for that ...
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1answer
182 views

Infinite $p$-extension contains $\mathbb{Z}_p$-extension

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$? My feeling is "yes", but I'm ...
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195 views

Is there an analogue of outer Space to study outer automorphisms of free pro-$p$ groups?

I would like to know if there is an analogue of Culler & Vogtmann's outer space to study outer automorphisms of free pro-$p$ groups. Perhaps an initial guess of such a space would be a moduli ...
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618 views

Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly infinite)...
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2answers
3k views

Galois group over $p$-adic numbers

Can one describe explicitly the Galois group $G=\operatorname{Gal}(\overline{\mathbb Q_p}/\mathbb Q_p)$? I only know the most basic stuff: unramified extensions of $\mathbb Q_p$ are equivalent to ...
3
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1answer
91 views

A dense subgroup with completion not isomorphic to the big (pro-p) group?

This is an (early) exercise from the book "Analytic Pro-p groups": (p.31, ex. 3(iii)) Give an example of a finitely generated pro-$p$ group $G$ and a dense subgroup $H$ of $G$, with $H$ finitely ...
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1answer
514 views

Conditions for a topological group to be a Lie group.

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157): Let $G$ be a locally compact group....
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79 views

Extending isomorphisms in the semi-simple case.

Is there some proposition saying how to extend an isomorphism of $k$-vector spaces where $k$ is a field of characteristic $p$ to an isomorphismus of $k[H]$-modules where $H$ is a group of order prime ...
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1answer
111 views

An induced exact sequence of $G$-modules for pro-$p$ group $G$

On p.64 of the book Cyclotomic Fields and Zeta Values by J. Coates and R. Sujatha: They seemed to have used the argument as follows: Let $G$ be a pro-$p$ group. If $0\rightarrow A\rightarrow B\...
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1answer
277 views

Importance and applications of profinite groups

Could someone tell me which is the importance and some applications of the profinite groups?
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1answer
1k views

Group representations over p-adic vector spaces

Recently I have found a need to learn more about p-adic group representations over a p-adic vector space. Generally, this motivates a study of representations $\left( V, \rho \right)$ for some group $...