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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 o …

Let be $G=F/F^p[F,F,F]$, with $F$ denotes the free group on $n$ generators. Then $G$ satisfies $Z(G) = \Phi (G)=G'$. If $x \in G-Z(G)$ then $C_G(x)= \langle x, Z(G) \rangle$ which is abelian as $C_ …

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 $ …

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 pe …

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 …

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 ge …

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 $G$ …

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 elem …

This answer is based on Holt's counter example. In a $p$-group of maximal class $G$, it is known that: $|G:G^p|=p^p$ and $\Omega_1(G)$ has either order $p^{p-1}$ or index $p$. Now we take $G$ of ma …

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 …

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: …

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 sub …

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 orde …

The classification of such groups is as difficult as the classification of all $p$-groups of maximal class. Note that, for the latter problem, beside the cases where $p=2,3$ that were settled by Blac …

Too long for a comment. I note first that I made an attempt to reduce the problem to the case where $K$ is normal, but it turned out to be false; I'm thankful to Ian Agol for his discussion. The cas …