# Tagged Questions

**1**

vote

**0**answers

101 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 ...

**8**

votes

**1**answer

172 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**

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**0**answers

69 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**

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**0**answers

44 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

**0**answers

65 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

**3**answers

322 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 ...

**1**

vote

**1**answer

64 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**

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**0**answers

46 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

108 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 ...

**9**

votes

**3**answers

454 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

**1**answer

105 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

**1**answer

152 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

114 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 $G$ ...

**4**

votes

**1**answer

150 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 ...

**21**

votes

**1**answer

753 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

**2**answers

393 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

**2**answers

418 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} ...

**3**

votes

**3**answers

497 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 ...

**4**

votes

**2**answers

464 views

### Center of finite metabelian p-groups

$\DeclareMathOperator\rk{rk}$
Let $G$ be a finite metabelian $p$-group, i.e. the commutator subgroup $G'$ of $G$ is abelian. Then I ask myself under which conditions does the following hold:
...