  Let $k$ be an algebraically closed field of characteristic $p > 0$, and let
$G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$.
Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding
finite Chevalley group. In this paper we investigate the structure of the first
cohomology group $H^1(G(\mathbb{F}_q),L(\lambda))$ where $L(\lambda)$ is the
simple $G$-module of highest weight $\lambda$. Under certain very mild
conditions on $p$ and $q$, we are able to completely describe the first
cohomology group when $\lambda$ is less than or equal to a fundamental dominant
weight. In particular, in the cases we consider, we show that the first
cohomology group has dimension at most one. Our calculations significantly
extend, and provide new proofs for, earlier results of Cline, Parshall, Scott,
and Jones, who considered the special case when $\lambda$ is a minimal nonzero
dominant weight.
