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Let k be an algebraically closed field of characteristic p &#62 0,and let G be a simple, simply connected algebraic group defined over Fp.Given r ≥ 1, set q = pr, and let G(Fq) be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group H1(G(Fq), L(λ)), where L(λ) is the simple G-module of highest weight λ. Under certain very mild conditions on p and q, we are able to completely describe the first cohomology group when λ 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 λ is a minimal non-zero dominant weight. [ABSTRACT FROM AUTHOR]