• Bulletin of the Korean Mathematical Society
• /
• v.53 no.3
• /
• pp.657-665
• /
• 2016

A general notion of ${\chi}$-transitive groups was introduced by C. Delizia et al. in [6], where ${\chi}$ is a class of groups. In [5], 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 ${\mid}E^2(G){\mid}$.