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.
Questions tagged [pro-p-groups]
42 questions
28
votes
2 answers
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
- 22,450
12
votes
1 answer
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…
Joshua Seaton
- 2,258
8
votes
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
- 81
8
votes
1 answer
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
- 3,317
7
votes
1 answer
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…
BIS HD
- 2,668
6
votes
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'.…
Giacomo Parolin
- 346
5
votes
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
- 1,017
5
votes
1 answer
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 $k$…
BIS HD
- 2,668
5
votes
0 answers
Induced topology on subgroup of free pro-$p$ group
I am trying to determine whether the following is true:
Let $H$ be a closed subgroup of a finitely generated free pro-$p$
group $F$. Does $F$ induce on $H$ the full pro-$p$ topology?
Stated in another way, is it true that every $p$-power index…
Ettore
- 711
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…
N.B.
- 2,149
5
votes
0 answers
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
- 73
4
votes
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
- 2,668
4
votes
1 answer
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
- 553
4
votes
1 answer
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 finite quotient is a $p$-group.
As my first attempt,…
Ariadne
- 128
3
votes
0 answers
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) with Galois group $G$. Now I denote by $G_1$…
BIS HD
- 2,668