Groups of prime power order and their nonabelian tensor squares

We prove that the nonabelian tensor square of a powerful p-group is again a powerful p-group. Furthermore, If G is powerful, then the exponent of G ⊕ G divides the exponent of G. New bounds for the exponent, rank, and order of various homological functors of a given finite p-group are obtained. In particular, we improve the bound for the order of the Schur multiplier of a given finite p-group obtained by Lubotzky and Mann.