  In this article we generalize the injectivity of the push-forward
homomorphism at the level of Chow groups, induced by the closed embedding of
$\Sym^m C$ into $\Sym^n C$ for $m\leq n$, where $C$ is a smooth projective
curve, to symmetric powers of a smooth projective variety of higher dimension.
We also prove the analog of this theorem for product of symmetric powers of
smooth projective varieties. As an application we prove the injectivity of the
push-forward homomorphism at the level of Chow groups, induced by the closed
embedding of self product of theta divisor into the self product of the
Jacobian of a smooth projective curve.
