All Questions
Tagged with gr.group-theory p-groups
121 questions
-2
votes
1
answer
162
views
Number of subgroups of a group of orders $p^3$ [closed]
Let $p$ be a prime number. Is there a formula for the number of subgroups of
$$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p^2\mathbb{Z}$$
$$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}\times\...
- 39
1
vote
0
answers
239
views
Cyclic subgroups of finite $p$-groups
Let $G$ be a finite non-Dedekind $p$-group with non-cyclic center, where $p$ is an odd prime.
By $[\langle x\rangle]_G=\{g^{-1}\langle x\rangle g\ |\ g\in G\},$
I mean the conjugacy class of the ...
- 487
0
votes
2
answers
956
views
Existence of a cyclic non-normal subgroup in a $p$-group
Let $G$ be a finite non-abelian $p$-group, where $p$ is an odd prime,
$N$ be a normal subgroup of $G$ of order $p$, where $\frac{G}{N}$ is non-abelian.
Does there exist an element $g\in G$ such that ...
- 487
6
votes
1
answer
854
views
Extra special p-groups
Let $P$ be an infinite extra special $p$-group for some prime $p$, namely, $Z(P)=P'=\Phi(P)$ and $P/Z(P)$ is infinite elementary abelian.
Let $C$ be a Prufer $q$-group for some prime $q\neq p$.
...
- 137
2
votes
1
answer
860
views
On $ p $-groups with at least one element of order $ p^{2} $
Let $G$ be a finite non-abelian $p$-group such that $G$ contains at least an element of order $p^2$ and for every nontrivial normal subgroup $N$, $G/N$ has not any elements of order p^2 and G/Z(G) is ...
- 21
3
votes
1
answer
248
views
p-groups with maximal class subgroup
Suppose $G$ is a finite non-abelian p-group of nilpotent class $c$. Is there a subgroup $H$ of nilpotent class $c$ and size $p^{c+1}$?
If this is not true, is it possible to add some additional ...
- 67
0
votes
1
answer
198
views
p-groups with special property on its centralizers
Thanks for any help or comment.
Suppose $G$ is a finite non-abelian p-group. Suppose $G$ has a proper non-abelian subgroup $M$ such that for every non-central element $x\in M$, $C_G(x)\subseteq M$. ...
- 71
32
votes
3
answers
3k
views
Is there a nice explanation for this curious fact about cyclic subgroups?
Here's something that I noticed that quite surprised me.
Let $G$ be a finite abelian group. Consider the following expression.
$$
\nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H|
$$
It ...
- 6,290
3
votes
2
answers
268
views
p-groups as finite union of disjoint normal abelian subgroups
I was interested in knowing if groups with following property have been studied( like what can be said about structure of the group) :
"$G$ can be written as disjoint union of a given number of ...
- 31
7
votes
1
answer
166
views
Random pro-p groups via iterated uniformly random central extensions
Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group:
We want to construct an inverse system
$$\cdots \xrightarrow{\alpha_i} G_i \...
- 401
9
votes
1
answer
282
views
p-groups such that the center is contained in many cyclic subgroups
I'm looking for examples of $p$-groups $G$ with the following three properties:
the center of $G$ is $\mathbb{Z}/p$, and
$G^{\text{ab}} = (\mathbb{Z}/p)^n$ for some $n$, and
for every $g \in G$ whose ...
- 91
7
votes
1
answer
370
views
Is $[729,57]$ a Sylow $3$-subgroup of some well-known group?
Let $G$ be the group $[729,57]$, using GAP's notation. I have so far two descriptions of the group:
a presentation
an embedding (not surjective!) of the group into a Sylow $3$-subgroup of the unit ...
user76083
4
votes
1
answer
360
views
Representations of p-groups where 1 is never an eigenvalue
Fix some $n \geq 1$ and some prime $p$. I'm looking for finite $p$-groups $G$ and finite-dimensional complex representations $V$ of $G$ with the following two properties:
The abelianization of $G$ ...
- 41
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
2
votes
1
answer
834
views
p-groups with unique normal minimal subgroup
Have $p$-groups with a unique normal minimal subgroup been classified? Is there any article on the subject?
- 21
6
votes
1
answer
312
views
Maximal cyclic quotient of a $p$-group
Let $G$ be a finite abelian $p$-group, $p$ a prime. I say that a pair $(G',\varphi)$ is a maximal cyclic quotient (please excuse me if this definition already exists and refers to a different concept) ...
- 509
4
votes
0
answers
175
views
Is the Tensor/Exterior square $G\otimes G$ or $G\wedge G$ of infinite p-group also a p-group?
Let $G$ be an infinite countable p-group. Is it true that $G\otimes G$ or $G\wedge G$ are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that $G=[G,G]$, and ...
- 647
1
vote
1
answer
192
views
Involutive automorphisms of a finite abelian p-group
First, let $A$ be a finitely generated free abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A_i$, where ...
- 14.1k
5
votes
0
answers
299
views
A class 3 group of order 243
Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a (...
user76083
0
votes
2
answers
403
views
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
9
votes
8
answers
5k
views
classification of $p$-groups
I have two questions regarding to $p$-groups.
A $p$-group $G$ is said to be extraspecial of $G'=Z(G)$ has order $p$. Hence extraspecial groups are examples of $p$-groups with cyclic center. Of ...
- 2,508
5
votes
1
answer
221
views
Finite solvable groups are generated by a nilpotent subgroup + K elements?
Is there a constant $K \in \mathbb{N}$ such that for every finite solvable group $G$, there exists a nilpotent subgroup $N \leq G$, and a subset $S \subseteq G$ with $|S| \leq K$, and $\langle N,S\...
- 11.3k
4
votes
1
answer
195
views
Finite p-groups and their fibered products
Is every finite $p$-group an epimorphic image of a fibered product of two finite $p$-groups which can be generated by $2$ elements?
- 11.3k
6
votes
0
answers
200
views
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
views
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
1
vote
3
answers
278
views
p-groups and 2-generated abelian images
Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?
- 11.3k
6
votes
0
answers
447
views
An example of a simple infinite 2-group
I've asked this question before on Mathematics, and they suggested me to ask here (Link).
Is there an example of a simple infinite $2$-group?
Informations
If a $2$-group is Artinian I know that it ...
- 599
1
vote
1
answer
308
views
Subgroups of the union of conjugates
This question is an attempt to find a version of this question with a positive answer. In this question, we ask if one can find big subgroups in the union of conjugtes of small subgroups of finite $p$-...
- 11.3k
2
votes
1
answer
216
views
Union of conjugates in p-groups
Fix a prime number $p$. Is there a sequence $\{a_k\}_{k \in \mathbb{N}}$ of real numbers with $$\lim_{k \to \infty} a_k = 0$$
such that for any finite $p$-group $G$, and any subgroup $H \leq G$ with ...
- 11.3k
8
votes
2
answers
295
views
Is the free abstract group residually of rank d > 2?
Let $d \geq 2$ be an integer, and let $\mathcal{F}_d$ be the family of finite groups such that $G \in \mathcal{F}_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements.
Is ...
- 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
9
votes
2
answers
380
views
Which finite p-groups occur as commutators of finite p-groups?
Let $p$ be a prime number. For which finite $p$-groups $H$ is there a finite $p$-group $G$ such that $[G,G] \cong H$?
- 11.3k
4
votes
0
answers
193
views
On a problem of Berkovich
What is the real history of the following problem proposed by Berkovich [Y. Berkovich, Z.Janko, Groups of prime power order. Volume 2, Expositions in Mathematics, 56, Walter de Gruyter, New York, 2011]...
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
13
votes
1
answer
455
views
Variety of nilpotent Lie algebras or $p$-groups
Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either:
1) Let $\mathcal{L}$ ...
- 63.9k
1
vote
0
answers
134
views
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
views
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
6
votes
3
answers
964
views
Union of conjugates of a subgroup
Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special ...
- 11.3k
0
votes
1
answer
326
views
A question on direct limits of finite $p$-groups
Where can we find a well developed material on direct limits of finite $p$-groups?
For instance, is there a characterization of such groups, which have a finite rank (that is every subgroup can be ...
2
votes
1
answer
105
views
A characterization of almost relatively free, finite $p$-groups
Let $G$ be a finite minimally $d$-generated $p$-group.
If $G$ is relatively free, that is $G$ is a quotient of the free group $F$ on $d$ generators by a fully invariant subgroup of $F$, then the ...
4
votes
2
answers
659
views
A question on $p$-central $p$-groups
Let $p$ be a fixed prime. A group $G$ is termed $p$-central if every element of order $p$ in $G$ lies in the center.
Having a finite $p$-group $G$ of rank $k$ (the least integer, such that every ...
2
votes
0
answers
202
views
Two $p$-groups whose automorphism groups have isomorphic Sylow $p$-subgroups
Fix a prime $p$, and let $M$ be the unique nonabelian group of order $p^3$ and exponent $p$. Let us denote by $E_n$ the elementary abelian group of rank $n$.
Is it true that $\operatorname{Aut}(M \...
8
votes
2
answers
470
views
Uniform-in-p classification* of p-groups of order p^n for each fixed n?
To what extent is there/can there be a description that is uniform in p (for p sufficiently large) of the p-groups of order $p^n$, for each fixed n?
Note 1: I used the word "description" rather than ...
- 3,230
10
votes
4
answers
1k
views
Classification of automorphism groups of groups of order $p^4$
For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of p-...
- 754
4
votes
2
answers
432
views
Index of agemo subgroups in $p$-groups
Having a finite $p$-group $G$ ($p$ odd). we denote by $\Omega_1(G)$ the subgroup generated by all the elements of $G$ of order dividing $p$.
Is there an example of such a group $G$, such that $|G:...
1
vote
1
answer
306
views
A finite $p$-group with certain properties
Is there a finite $p$-group $G$ such that :
(a) $G= \langle A,x,y \rangle$, with $G/Z(G)$ has exponent $p$, $A$ is a maximal abelian normal subgroup of $G$, and $G/A$ has order $p^2$ (thus it is ...
2
votes
0
answers
130
views
Non left $k$-Engel elements in a nilpoent group always generate this group
Given a finite nilpotent group $G$ and let us denote by $L_n(G)$ the set of left $n$-Engel elements in $G$.
Assume that $n$ is the smallest positive integer such that $L_n(G)=G$.
Is it true that $G$ ...
1
vote
0
answers
134
views
The number of $p$-groups of order $\leq p^n$ that split over a normal abelian subgroup
How can one estimate the number of $p$-groups of order $\leq p^n$ that split over a normal abelian subgroup?
Moreover, let $s(n,p)$ be the number of such groups, and let $f(n,p)$ denotes the number ...