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### 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 ...

**13**

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282 views

### p-groups as rational points of unipotent groups

Is it true that every finite p-group can be realized as the group of rational points over $\mathbb{F_p}$ of some connected unipotent algebraic group defined over $\mathbb{F_p}$? For abelian p-groups, ...

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114 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, ...

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50 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 ...

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123 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 ...

**2**

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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 $G$ ...

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79 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 ...

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45 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 ...

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107 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 ...

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62 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 ...

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46 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 ...

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118 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 ...

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74 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 $\Phi \cap N = \{1\}$?

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66 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 $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$

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102 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 ...