On generalized $2_{\otimes }$-auto Engel subgroups

Abstract

The notion of $2_{\otimes }$-auto Engel subgroups was studied in Moghaddam and Sadeghifard (Bull Korean Math Soc 52:1097–1105, 2015). Let N be a characteristic subgroup of a group G including $G^{\prime }$ and $Aut^{N} (G)$ be the set of all automorphisms of group G which induces the identity on G / N. In the present paper, through the use of $Aut^{N} (G)$ we introduce the notion of $2_{\otimes }$-N-auto Engel elements in group G, which is a generalization of $2_{\otimes }$-auto Engel elements of G. In particular we show that the set of these elements is a characteristic subgroup of the group. So every characteristic subgroup N of G containing $G'$, induces a corresponding characteristic subgroup $A^{N}R^{\otimes }_{2}(G)$ in G.