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Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U ) with coefficients in k.

Let G(Fq) be a finite Chevalley group defined over the field of q = p r elements, and k be an algebraically closed field of characteristic p > 0. A fundamental open and elusive problem has been the computation of the cohomology ring H • (G(Fq), k). In this paper we determine initial vanishing ranges which improves upon known results. For root systems of type An and Cn, the first non-trivial cohomology classes are determined when p is larger than the Coxeter number (larger than twice the Coxeter number for type An

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