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4
votes
1answer
155 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 ...
4
votes
1answer
159 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?
1
vote
0answers
67 views

finite p-group subgroup of infinite p-group

is there any finite p-group G that is subgroup or minimal/maximal subgroup of infinite p-group H? if yes what is the limits? can this happen with different p's? i'm more interested in being maximal ...
3
votes
0answers
83 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 $n$ ...
2
votes
0answers
104 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 ...
1
vote
1answer
102 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?
4
votes
1answer
160 views

Can every finite abelian $p$-group with duality pairing be written as cokernel of a symmetric matrix over the $p$-adic integers?

Let $G$ be a finite abelian $p$-group (where $p$ is a prime). Suppose there exists a symmetric bilinear map $\delta\colon G\times G\to \mathbb{Q}/\mathbb{Z}$ such that the induced map $g\to\langle ...
6
votes
0answers
157 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 ...
1
vote
0answers
87 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 ...
2
votes
1answer
175 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 ...
7
votes
2answers
231 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 ...
1
vote
0answers
53 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 ...
7
votes
1answer
149 views

Which finite p-groups occur as commutators of finite p-groups?

Let $p$ be a prime number. For which finite $p$-groups $H$ there is a finite $p$-group $G$ such that $[G,G] \cong H$?
3
votes
0answers
131 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, ...
1
vote
0answers
117 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 ...
1
vote
0answers
70 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 ...
10
votes
1answer
235 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 $\mathcal{L}$ ...
0
votes
0answers
80 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 $\Phi \cap N = \{1\}$?
1
vote
0answers
52 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 ...
0
votes
0answers
69 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 $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
6
votes
3answers
377 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 ...
0
votes
1answer
149 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 ...
1
vote
1answer
73 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 ...
2
votes
0answers
58 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
0answers
136 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 ...
7
votes
2answers
226 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 ...
9
votes
3answers
493 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 ...
1
vote
1answer
108 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 ...
0
votes
1answer
158 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
0answers
115 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
0answers
121 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 ...
2
votes
1answer
355 views

p-group with abelian centralizer

I will be so thankful if someone helps me with the following question. There exists finite non-abelian p-groups G (except non-abelian groups of order $p^3$) with the following properties: all ...
4
votes
1answer
156 views

Number of generators of the automorphism group of an abelian group

Let $G$ be a finite abelian $p$-group. What is known about the minimal number of generators of a $p$-sylow of $Aut(G)$? is it bounded in terms of $d(G)$ the minimal number of generators of $G$ (and ...
3
votes
2answers
310 views

Normal abelian subgroups in p-groups

Given a group $G$, we denote by $T(G)$ the subgroup generated by all (maximal) normal abelian subgroups of $G$. Let define the series $(T_i(G))$ by $T_0(G)=1$ and $T_{i+1}(G)/T_i(G)=T(G/T_i(G)$, and ...
0
votes
0answers
105 views

Centralizers of elementary abelian subgroups of $p$-groups

Let $P$ be a $p$-group. It is known that if $E$ is a maximal elementary abelian subgroup of rank 2 in $P$, then $C_P(E)/E$ is cyclic where $C_P(E)$ denotes the centralizer of $E$ in $P$. This is ...
2
votes
2answers
237 views

P-group with abelian centralzer

I will be so thankful if someone help me about the following question. I need to know the presentation of a (if it is possible) family of finite non-abelian $p$-group $G$ with the follwing properties: ...
6
votes
3answers
358 views

Characters of p-groups

Berkovich mentioned the following result of Mann in his book on p-groups: The number of nonlinear irreducible characters of given degree in a p-group is divided by p-1. Do you know any reference for ...
1
vote
1answer
218 views

Normal subgroups In a p-group [Reference?]

Dear Experts, I'm a graduate student, dealing with group-theory. In my current research, I used the bound "Alexander Gruber" wrote about in this post: See Here (Actually, I have just found out ...
4
votes
1answer
343 views

Generators of p-groups

Let $G$ be a finite $p$-group. Since we can embed $Z_2(G)/Z(G)$ in $Hom(G,Z(G))$, we have $d_2 \leq d(G)d(Z(G))$; where $d_2(G)=d(Z_2(G)/Z(G))$ and $d(G)$ denotes the minimal number of generators of ...
4
votes
2answers
358 views

p-group with large center

Is there any characterization for $p$-groups of order greater than $p^3$ which center has index $p^2$? (One group whit this property if $M(p^n)$)
22
votes
1answer
778 views

Richness of the subgroup structure of p-groups

Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest number such that there is a group of order $p^{f_p(n)}$ which all groups of order $p^n$ embed into. What is the asymptotic growth ...
1
vote
2answers
420 views

finite abelian p-groups with solvable automorphism group

Let $G$ be an abelian (not elementary) finite $p$-group. In what conditions the automorphism group of $G$ is solvable?
7
votes
2answers
436 views

Automorphism Group of a p-group : Looking for a Reference

In the following post by DavidLHarden : See Here He quoted the following claim: "There is a theorem that says that if $p$ is a prime and $|G|=p^n $ , then $|AutG| $ divides $ \Pi_{k=0}^{n-1} ...
4
votes
3answers
2k views

Number of Normal subgroups In a p-Group

Dear all, Does someone know of any paper/method that enables us counting/estimating the number of normal subgroups of some p-group of order $p ^n $ ($ n$ is some natural number ? ) . Is there anyway ...
2
votes
1answer
140 views

Hall algebra for non-abelian p-groups ?

According to WP article on Hall algebras one counts the number of abelian subgroups in abelian group with fixed type of subgroup, group, quotient. Two things are claimed: 1) These numbers are ...
2
votes
3answers
985 views

Center of p-groups

Can one show that any abelian $p$-group (not necessarily finite) is the center of a $p$-group and of index $p$?
3
votes
3answers
624 views

Representation theory of p-groups in particular upper tringular matrices over F_p

Finite p-groups - have p^n elements by definition. According to WP there is rich structure theory. Question: How far is representation theory of p-groups is understood? In case this question is too ...
7
votes
1answer
356 views

Maximal subgroups of a certain finite 2-group

The following came up in a problem on reconstruction of digraphs. I determined enough about the answer to satisfy the application completely, but still I am curious to know what the complete solution ...
7
votes
2answers
713 views

Representation theory of a finite p-group over a field of characteristic p: dim of invariants =1 => dim of coinvariants = 1?

I am trying to understand the proof of Proposition 4 in S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. Volume 39 (1970), 141-148. The PDF is available here: ...
3
votes
2answers
573 views

Maximum value of the number of conjugacy classes of nonabelian p-groups with an abelian subgroup of index p

It is known that if $G$ is a nonabelian $p$-group of order $p^n$, with an abelian subgroup of index $p$, then the number $k(G)$ of conjugacy classes of $G$ can be as large as $p^{n-1} + p^{n-2} - ...