  We prove that the product of a subset and a normal subset inside any finite
simple non-abelian group $G$ grows rapidly. More precisely, if $A$ and $B$ are
two subsets with $B$ normal and neither of them is too large inside $G$, then
$|AB| \geq |A||B|^{1-\epsilon}$ where $\epsilon>0$ can be taken arbitrarily
small. This is a somewhat surprising strengthening of a theorem of Liebeck,
Schul, Shalev.
