$2$-Engelizer subgroup of a $2$-Engel transitive groups
Bull. Korean Math. Soc. 2016 Vol. 53, No. 3, 657-665
Published online May 31, 2016
Mohammad Reza R. Moghaddam and Mohammad Amin Rostamyari
Ferdowsi University of Mashhad, Ferdowsi University of Mashhad
Abstract : 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 $|E^2(G)|$.
Keywords : $2$-ET group, $\rm{CSE}^2$-group, residually $\chi$-group, fully residually $\chi$-group, $2$-Engelizer subgroup
MSC numbers : Primary 20F19, 20E06, 20B08; Secondary 20F99, 20E70
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