# 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.

42 questions
Filter by
Sorted by
Tagged with
0answers
24 views

### Cardinality of a elementary abelian pro-$p$ group

Let $G$ be an elementary abelian pro-$p$ group. Then we have that $$G=\prod\limits_{\mathfrak{m}}C_p$$ where $\mathfrak{m}$ is a cardinal. We have this as a direct consequence of Theorem 4.3.8. from ...
0answers
28 views

### The Finiteness of H^n(H, Z/pZ) implies finiteness of H^n(U, Z/pZ) for any open subgroup U of H

On Page 175 of J.Neukirch et al.'s book Cohomology of Number Fields, it was remarked: If $H$ is a pro-$p$-group, then this is already true if $H^n(H, \mathbb{Z}/p\mathbb{Z})$ is finite ($n = cd_{p}H$)...
0answers
14 views

### $\mathcal{N}_p$-words and free pro-$p$ groups

Let $\omega$ be a word such that every finitely generated pro-$p$ group $H$ satisfies $H/\overline{\omega(H)}$ nilpotent-by-finite. Choose some free pro-$p$ group $H$ generated by $x_1,...,x_d,y$. Why ...
1answer
64 views

1answer
88 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 ...
1answer
122 views

1answer
149 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 ...
0answers
30 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, $R$...
0answers
79 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 ...
1answer
196 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,...
1answer
75 views

0answers
49 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 $H$. ...
2answers
539 views

2answers
83 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 ...
1answer
488 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 ...
1answer
411 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 $G$...
1answer
340 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 ...
1answer
191 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 ...
0answers
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 ...
0answers
700 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)...
2answers
4k 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 ...
1answer
108 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 ...
1answer
534 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....
0answers
80 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 ...
1answer
111 views