  Let $G$ be a finitely generated group. We prove that the $n$-fold tensor
product $G^{\otimes n}$ is finite (resp. polycyclic) if and only $G$ is finite
(resp. polycyclic). Further, assuming that $G$ is finitely presented, we show
that $G^{\otimes n}$ is finitely presented if and only if $\gamma_n(G)$ is
finitely presented. We also examine some finiteness conditions for the
non-abelian tensor product of groups.
