On some subgroups associated with the tensor square of a group

Abstract

‎In this paper we present some results about subgroup which is‎ ‎generalization of the subgroup $R_{2}^{otimes}(G)={ain‎ ‎G|[a,g]otimes g=1_{otimes},forall gin G}$ of right‎ ‎$2_{otimes}$-Engel elements of a given group $G$‎. ‎If $p$ is an‎ ‎odd prime‎, ‎then with the help of these results‎, ‎we obtain some‎ ‎results about tensor squares of p-groups satisfying the law‎ ‎$[x,g,y]otimes g=1_{otimes}$‎, ‎for all $x‎, ‎g‎, ‎yin G$‎. ‎In‎ ‎particular p-groups satisfying the law $[x,g,y]otimes‎ ‎g=1_{otimes}$ have abelian tensor squares‎. ‎Moreover‎, ‎we can‎ ‎determine tensor squares of two-generator p-groups of class three‎ ‎satisfying the law $[x,g,y]otimes g=1_{otimes}$‎.