Document Type: Research Paper
Department of Mathematics (University of Salerno), Italy - Department of Mathematics and Statistic (University of the Basque Country), Spain
Dipartimento di Matematica, Università di Salerno - Italy
Let $\Gamma$ be the first Grigorchuk group. According to a result of Bar\-thol\-di, the only left Engel elements of $\Gamma$ are the involutions. This implies that the set of left Engel elements of $\Gamma$ is not a subgroup. The natural question arises whether this is also the case for the sets of bounded left Engel elements, right Engel elements and bounded right Engel elements of $\Gamma$. Motivated by this, we prove that these three subsets of $\Gamma$ coincide with the identity subgroup.