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
3
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2
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Sylow subgroups of $\mathrm{SL}_2(\mathbb{Z}_p)$ and $\mathrm{SL}_2(\mathbb{F}_p)$
Let $p,l$ be odd primes and $p\neq l$. I am interested in determining the $l$-Sylow subgroups of the groups $\mathrm{SL}_2(\mathbb{Z}_p)$ and $\mathrm{SL}_2(\mathbb{F}_p)$.
I have read that for $\...
- 1,270
3
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1
answer
51
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Pro-$p$ Group with solvable subgroup of finite Index is solvable?
Let $p$ be a prime, and let $G$ be a pro-$p$-group. Suppose that $G$ contains a solvable subgroup $H$ of finite index. Is it true that $G$ is also solvable? Since any finite $p$-group is nilpotent (...
- 217
1
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0
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46
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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 ...
1
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0
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33
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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 $H^n(G, \...
- 483
0
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0
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78
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Cardinality of a elementary abelian pro-$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 ...
- 523
1
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1
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355
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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 $...
- 4,125
1
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0
answers
66
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Infinite pro-$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 $p$-...
- 3,581
1
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1
answer
188
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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 ...
- 1,998
1
vote
1
answer
69
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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. (...
- 984
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0
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130
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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\...
- 346
1
vote
1
answer
355
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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 ...
- 2,119
5
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1
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191
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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]\...
- 2,119
0
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1
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104
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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 ...
- 549
2
votes
2
answers
87
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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 ...
- 4,125
0
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1
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74
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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 ...
- 4,125
0
votes
1
answer
123
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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 ...
- 4,125
2
votes
0
answers
285
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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 $\...
- 1,826
3
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1
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317
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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 ...
- 73
1
vote
0
answers
79
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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, $R$...
- 183
0
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0
answers
166
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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 ...
- 8,441
1
vote
1
answer
301
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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,...
- 347
2
votes
1
answer
85
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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+\...
- 913
4
votes
1
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439
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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 ...
- 128
3
votes
2
answers
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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 $\...
- 1,933
5
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0
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58
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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 $H$.
...
- 73
2
votes
2
answers
807
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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 $|...
- 4,350
2
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0
answers
73
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Proof that a particular subgroup is proper
I've been stuck on this for a long time ... I'm reading a textbook which simply states "this subgroup is proper" but it doesn't make sense to me.
Context: I have a pro-$p$ group $G$, which just means ...
- 3,275
8
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1
answer
969
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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(...
- 3,275
5
votes
2
answers
100
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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 ...
- 1,017
5
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1
answer
759
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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 ...
- 2,658
1
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1
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842
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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 $G$...
- 2,658
4
votes
1
answer
452
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$p$-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 ...
- 2,658
7
votes
1
answer
267
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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 ...
- 2,658
3
votes
0
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958
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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)...
- 2,658
28
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2
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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 ...
- 22.2k
3
votes
1
answer
154
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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 ...
- 31
12
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1
answer
633
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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....
- 2,238
1
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0
answers
83
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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 ...
- 2,658
4
votes
1
answer
154
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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\...
- 553
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1
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419
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Importance and applications of profinite groups
Could someone tell me which is the importance and some applications of the profinite groups?
- 327
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1
answer
1k
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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 $...
- 81