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.

41 questions
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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 extensions of $\mathbb F_p$, so the Galois group of…
user8268
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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. Then (Gleason-Montgomery-Zippin-Yamabe) G is…
8
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1 answer

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 $G$ where $V$ is a vector space over…
jdmorgan
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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(\mathbf{Z}_p) : g\equiv \text{id} \, (\text{mod…
Ehsaan
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7
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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 not sure if we need the extension $K/k$ to be…
6
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0 answers

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 $x\in G$ s.t. $$ K_x(G):=\{[x,g]\mid g\in G \}=G'.…
5
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2 answers

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 what I have so for. The order of $a$ is $p$ since…
abe
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5
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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 space of minimal free actions of a fixed free pro-$p$…
5
votes
1 answer

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 $[G,G]\leq G^4$. Why this discrepancy between odd and…
5
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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 any open uniform subgroup of $G$…
Lucas
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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 $H$. It is enough to show that each normal subgroup $N$…
Milford
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4
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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 of $k$ by $\mathfrak{p}$. For a number field $k$…
4
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1 answer

$p$-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 fact? In the book "Analytic Pro-$p$ Groups" of…
BIS HD
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4
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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 $d$. (Here, the book assume $\mathrm{d}(G) = d$). I'm in…
Lucas
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4
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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 $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is an exact sequence of…
ksj03
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