  Let $k$ be an algebraically closed field of characteristic $p > 0$, and let
$G$ be a simple simply-connected algebraic group over $k$ that is defined and
split over the prime field $\mathbb{F}_p$. In this paper we investigate
situations where the dimension of a rational cohomology group for $G$ can be
bounded by a constant times the dimension of the coefficient module. We then
demonstrate how our results can be applied to obtain effective bounds on the
first cohomology of the symmetric group. We also show how, for finite Chevalley
groups, our methods permit significant improvements over previous estimates for
the dimensions of second cohomology groups.
