All Questions
Tagged with pro-p-groups profinite-groups
19 questions
3
votes
2
answers
165
views
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 $\...
- 1,270
3
votes
1
answer
51
views
Pro-$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
vote
0
answers
33
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 $H^n(G, \...
- 483
0
votes
0
answers
78
views
Cardinality of a elementary abelian pro-$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 ...
- 523
1
vote
1
answer
355
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 $...
- 4,125
1
vote
1
answer
188
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 ...
- 1,998
1
vote
1
answer
69
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. (...
- 984
1
vote
1
answer
355
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 ...
- 2,119
5
votes
1
answer
191
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]\...
- 2,119
0
votes
1
answer
104
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 ...
- 549
2
votes
0
answers
285
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 $\...
- 1,826
0
votes
0
answers
166
views
Example Powerful Pro-$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 ...
- 8,441
2
votes
1
answer
85
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+\...
- 913
5
votes
0
answers
58
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 $H$.
...
- 73
2
votes
2
answers
807
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 $|...
- 4,350
8
votes
1
answer
969
views
Nontrivial examples of pro-$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. $\Gamma_1:=\{g\in \text{SL}_n(...
- 3,275
1
vote
1
answer
842
views
Cohomological ($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 $G$...
- 2,658
3
votes
1
answer
154
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 ...
- 31
3
votes
1
answer
419
views
Importance and applications of profinite groups
Could someone tell me which is the importance and some applications of the profinite groups?
- 327