All Questions
Tagged with profinite-groups p-groups
16 questions
3
votes
1
answer
133
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Any Sylow pro-$p$ subgroup of a topologically finitely generated profinite group is also topologically finitely generated?
It's proved in Oltikar and Ribes - On prosupersolvable groups that any Sylow pro-$p$ subgroup of a topologically finitely generated prosupersolvable group is also topologically finitely generated. It ...
- 667
6
votes
2
answers
188
views
Agemo-of-agemo inclusions for p-groups
For a finite $p$-group $G$, let $\mho_i(G)$ denote the subgroup generated by $p^i$-powers of elements of $G$.
It is well-known that $\mho_i(\mho_j(G))$ can differ from $\mho_j(\mho_i(G))$ and from $\...
- 2,519
0
votes
0
answers
54
views
Existence of maximal topologically characteristic subgroup of infinite index of pro-$p$ groups
Let $G$ be a topologically finitely generated infinite pro-$p$ group. Suppose that $G$ is not just-infinite. Does the group $G$ always have a maximal topologically characteristic subgroup of infinite ...
- 667
1
vote
1
answer
102
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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. I asked this question in math....
- 243
3
votes
1
answer
348
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For a pro-p, profinite group, abelianization being finitely generated is the same as being topologically finitely generated
I remember reading (without proof) that for $\Gamma$ a profinite, pro-$p$ group, the following are equivalent:
1) Every open subgroup $\Gamma_0$ is topologically finitely generated.
2) The ...
- 7,736
7
votes
1
answer
166
views
Random pro-p groups via iterated uniformly random central extensions
Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group:
We want to construct an inverse system
$$\cdots \xrightarrow{\alpha_i} G_i \...
- 401
0
votes
1
answer
230
views
Uniform pro-p groups as a semi-direct product
Let $G$ a finitely generated uniform pro-$p$ group. Then $G/[G,G]$ is abelian and so it is of the form $\mathbb{Z}_p^r\times T$ for some integer $r$ and finite $p$-group $T$. Therefore, $[G,G]$ is ...
- 621
0
votes
2
answers
403
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Is the Frattini subgroup of a f.g virtually pro-p group open?
Let $G$ be a finitely generated profinite group, and $p$ a prime number. Suppose that there exists some open pro-$p$ subgroup $H \leq_o G$. Must $G$ have only finitely many maximal open subgroups?
...
- 11.3k
6
votes
0
answers
200
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Is there a Noetherian profinite group of infinite rank?
Is there a profinite group $G$ such that any closed subgroup $H \leq G$ is finitely generated, but there is no $n \in \mathbb{N}$ such that every closed subgroup of $G$ can be generated by at most $n$ ...
- 11.3k
6
votes
1
answer
719
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Torsion in profinite groups
Is there a finitely generated profinite group $G$ with a closed subgroup of infinite index $K \leq G$ such that for every $g \in G$ there exists some $n \in \mathbb{N}$ for which $g^n \in K$ ?
Can $G$...
- 11.3k
2
votes
0
answers
165
views
Exhausting a free pro-p group
Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$.
Let $p$...
- 11.3k
2
votes
1
answer
306
views
An epimorphism into a profinite group
Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...
- 11.3k
2
votes
0
answers
126
views
Bases for free pro-p groups
Let $p$ be a prime number, $F$ a free nonabelian finitely generated pro-$p$ group, $L \lhd_o F$ and $Y$ a basis for $L$ with $y \in Y$. Is there a basis $X$ for $F$ such that $y$ is in the abstract ...
- 11.3k
1
vote
0
answers
134
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Dense free subgroups
Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?
- 11.3k
1
vote
0
answers
166
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Lower central series in a free pro-p group
Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$.
Is ...
- 11.3k
1
vote
0
answers
82
views
A bound on the size of the center
Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
- 11.3k