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No, the rank of $G$ could be arbitrary large. Here is a counter-example that works for all primes $p$. Fix a positive integer $n$, and let $G$ be the direct sum of $n$ copies of $\mathbb{Z}/p^2\mathb …

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 …

Let $G$ be a group. Let us say that $G$ satisfies a generalized identity of degree $n$ if there exist $a_1,a_2,\dots a_n \in G$ such that $$x^{a_1}x^{a_2}\dots x^{a_n}=1,$$ for all $x\in G$. Assum …

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 …

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 …

For your second question the answer is yes. The bound follows from Schreier's inequality: if $\Phi(G)$ has index $p^d$, then it follows that $\Phi(G)$ can be generated by $p^d(d-1)+1$ elements. Note …

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 …

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 …

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: …

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 …

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 $G$ …

Let $G$ be a group of odd order. It is known that if every central automophism of $G$ acts trivially on the center, then $G$ is purely non-abelain, this amounts to saying that every central endomorph …

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 …