Are $p$-groups Engel groups? A group $G$ is termed Engel if whenever $x, y \in G$, there exists an integer $n$ (depending on $x$ and $y$ such that $[x,y,\dots,y]=1$, where $y$ occurs $n$ times.
Is it true that every infinite $p$-group is Engel? 
 A: No. There are infinite $p$-groups that are not Engel groups.
Baer (1957) showed that in groups in which every ascending chain of subgroups eventually stabilizes (“Noetherian groups”), the set of left Engel elements is the Fitting subgroup and the set of right Engel elements is the hypercenter. In particular, this generalizes Zorn's theorem (that a finite Engel group is nilpotent) to Noetherian groups (Noetherian+Engel implies nilpotent).
Since a Tarski monster is fairly trivially a Noetherian group (maximal chains of subgroups are $1 < \langle x \rangle < G$) the only left or right Engel element is the identity. Since a Tarski monster has exponent $p$, it is a $p$-group.
Be cautious that there are many open problems about Engel elements in general. Early results such as Baer's are very nice, but the later research is half theorem and half counterexample, as the behavior of Engel sets is difficult to predict.
This paper also has an important result in finite groups, the Baer-Suzuki theorem.


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*Baer, Reinhold.
“Engelsche elemente Noetherscher Gruppen.”
Math. Ann. 133 (1957), 256–270.
MR86815
DOI:10.1007/BF02547953
A: Well,  nilpotent groups are almost trivially $\;n-$ Engel, with $\;n=$ the group's nilpotency class. Since finite $\;p-$ groups are nilpotent then they are Engel groups.
About infinite $\;p-$groups I can't be sure, but I think in general the answer is negative since there are simple (infinite) $\;p-$groups...
