Let $p$ be a prime and let $F$ be a free group of rank $d\geq 1$.
Kostrikin [1] proved that the $d$-generated Burnside group $B=B(d,p)=F/F^p$
of exponent $p$ has a maximal finite quotient
$\overline{B}=\overline{B(d,p)}=B/N$ where $N$ is the intersection of all
the subgroups of $B$ of finite index. I understand that Kostrikin proved
that the nilpotency class $c(d,p)$ of $\overline{B(d,p)}$ has an *upper bound*.


**Question 1:** Are *lower bounds* known for $c(d,p)$?


**Question 2:** Is $c(d,p)\geq d$ for $d\leq 4$ and $p>d$?


**Question 3:** For which $n$ is the $n$th section of the lower exponent-$p$
central series for $\overline{B(d,p)}$ isomorphic to the $n$th Lie power
$L^n(V)$ where $V=({\mathbb F}\_p)^d$?


[1] Kostrikin, A. L: On Lie rings with Engel's condition. Dokl. Akad. Nauk SSSR 108, no. 4 (1956) 580-582.



Restricted Burnside Problem: Lower bound nilpotency class
For Questions 1, 2.
We know $c(d,2)=1$ for all $d$ and $c(1,3)=1$, $c(2,3)=2$ and $c(d,3)=3$ for all $d>2$.
So suppose if necessary $p\geq 5$. 


Take $G(d,p)$ to be the unitiriangular matrices of size $(d+1)\times (d+1)$ over the field of size $p$, where $p>d$. Then $G(d,p)$ is nilpotent of class $d$ and the exponent is $p$. The group can be generated by $d$ elements.


