Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
42 views

Exponent of a semi-direct product of nilpotent groups

It is well-known that in a semi-direct product of two groups $G = N\rtimes_\phi H$ the exponent of $G$ might be bigger than the lcm of the exponents of $N$ and $H$. It sometimes does not, I give an ...
Archistin Beedle's user avatar
1 vote
0 answers
28 views

Finiteness of $n$th Galois cohomology$H^n(U, \mathbb{F}_p)$ for open subgroups $U$ of a pro-$p$-group $G$

I am reading a book named 'cohomology of number fields' by Neukirch, Schmidt, Wingberg. Let $G$ be a pro-$p$-group. Suppose the $p$-cohomological dimension $cd_p G=n<\infty$. Suppose $H^n(G, \...
MiRi_NaE's user avatar
  • 473
0 votes
0 answers
66 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 ...
Gillyweeds's user avatar
1 vote
1 answer
289 views

Closed subgroup of a pro-p group

I want to prove the following proposition: Proposition. If $H$ is a closed subgroup of a pro-$p$ group $G$, then $H$ is pro-$p$ There is a result that maybe can be used in order to prove that. If $...
Lucas's user avatar
  • 4,103
1 vote
0 answers
61 views

Infinite pro-$p$ group of finite solvable length and finite coclass

I was reading about infinite pro-$p$ groups of finite coclass from the book "The Structure of Groups of Prime Power Order" by Leedham-Green and McKay. To recall, the coclass of a finite $p$-...
usermath's user avatar
  • 3,516
1 vote
1 answer
168 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 ...
curious math guy's user avatar
1 vote
1 answer
67 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. (...
debanjana's user avatar
  • 964
6 votes
0 answers
128 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\...
Giacomo Parolin's user avatar
1 vote
1 answer
329 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 ...
N.B.'s user avatar
  • 2,109
5 votes
1 answer
187 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]\...
N.B.'s user avatar
  • 2,109
0 votes
1 answer
73 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 ...
Khal's user avatar
  • 549
2 votes
2 answers
87 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 ...
Lucas's user avatar
  • 4,103
0 votes
1 answer
73 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 ...
Lucas's user avatar
  • 4,103
0 votes
1 answer
121 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 ...
Lucas's user avatar
  • 4,103
2 votes
0 answers
258 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 $\...
MathStudent's user avatar
  • 1,816
3 votes
1 answer
293 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 ...
user579852's user avatar
1 vote
0 answers
75 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, $R$...
gualterio's user avatar
  • 183
0 votes
0 answers
160 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 ...
user267839's user avatar
  • 7,109
1 vote
1 answer
284 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,...
Leo D's user avatar
  • 347
2 votes
1 answer
84 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+\...
Lukas's user avatar
  • 903
4 votes
1 answer
405 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 ...
Ariadne's user avatar
  • 128
3 votes
2 answers
918 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 $\...
Ashwin Iyengar's user avatar
5 votes
0 answers
57 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 $H$. ...
Milford's user avatar
  • 73
2 votes
2 answers
775 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 $|...
Alexander's user avatar
  • 4,302
2 votes
0 answers
69 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 ...
Ehsaan's user avatar
  • 3,245
8 votes
1 answer
929 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(...
Ehsaan's user avatar
  • 3,245
5 votes
2 answers
99 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 ...
abe's user avatar
  • 997
5 votes
1 answer
709 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 ...
BIS HD's user avatar
  • 2,663
1 vote
1 answer
759 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 $G$...
BIS HD's user avatar
  • 2,663
4 votes
1 answer
437 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 ...
BIS HD's user avatar
  • 2,663
7 votes
1 answer
247 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 ...
BIS HD's user avatar
  • 2,663
3 votes
0 answers
924 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)...
BIS HD's user avatar
  • 2,663
27 votes
2 answers
6k 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 ...
user8268's user avatar
  • 21.6k
3 votes
1 answer
147 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 ...
QuackQuack's user avatar
12 votes
1 answer
621 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....
Joshua Seaton's user avatar
1 vote
0 answers
83 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 ...
BIS HD's user avatar
  • 2,663
4 votes
1 answer
150 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\...
ksj03's user avatar
  • 553
3 votes
1 answer
395 views

Importance and applications of profinite groups

Could someone tell me which is the importance and some applications of the profinite groups?
Andres's user avatar
  • 327
8 votes
1 answer
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 $...
jdmorgan's user avatar