2-ENGELIZER SUBGROUP OF A 2-ENGEL TRANSITIVE GROUPS

A general notion of $\chi$-transitive groups was introduced by C. Delizia et al. in \cite{d}, where $\chi$ is a class of groups. In \cite{c}, Ciobanu, Fine and Rosenberger studied the relationship among the notions of conjugately separated abelian, commutative transitive and fully residually $\chi$-groups. In this article we study the concept of $2$-Engel transitive groups and among other results, its relationship with conjugately separated $2$-Engel and fully residually $\chi$-groups are established. We also introduce the notion of $2$-Engelizer of the element $x$ in $G$ and denote the set of all $2$-Engelizers in $G$ by $E^2(G)$. Then we construct the possible values of $|E^2(G)|$. KCI Citation Count: 0