  We calculate the smooth structure set of $S^p \times S^q$, $S(p, q)$, for $p,
q \geq 2$ and $p+q \geq 5$. As a consequence we show that in general $S(4j-1,
4k)$ cannot admit a group structure such that the smooth surgery exact sequence
is a long exact sequence of groups. We also show that the image of forgetful
map $F: S(4j, 4k) --> S^{Top}(4j, 4k)$ is not in general a subgroup of the
topological structure set.
