Non-Abelian tensor square and related constructions of p-groups

Let G be a group. We denote by ν ( G ) a certain extension of the non-Abelian tensor square [ G , G φ ] by G × G . We prove that if G is a finite potent p -group, then [ G , G φ ] and the k -th term of the lower central series γ k ( ν ( G ) ) are potently embedded in ν ( G ) (Theorem A). Moreover, we show that if G is a potent p -group, then the exponent exp ( ν ( G ) ) divides p · exp ( G ) (Theorem B). We also study the weak commutativity construction of powerful p -groups (Theorem C).