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2-ENGELIZER SUBGROUP OF A 2-ENGEL TRANSITIVE GROUPS
Moghaddam, Mohammad Reza R.,Rostamyari, Mohammad Amin Korean Mathematical Society 2016 대한수학회보 Vol.53 No.3
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 ${\mid}E^2(G){\mid}$.
$2$-Engelizer subgroup of a $2$-Engel transitive groups
Mohammad Reza R. Moghaddam,Mohammad Amin Rostamyari 대한수학회 2016 대한수학회보 Vol.53 No.3
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 $|E^2(G)|$.