On some series of a group related to the non-abelian tensor square of groups

Abstract

Let G be a group. We denote by ν(G) a certain extension of the non-abelian tensor square G⊗G by G×G. In this paper we prove that the derived subgroup ν(G)′ is a central product of three normal subgroups of ν(G), all isomorphic to the non-abelian tensor square G⊗G. As a consequence, we describe the structure of each term of the derived and lower central series of the group ν(G).