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 the 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 $[x,g,y]otimes g=1_{otimes}$