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Let G be a connected, simply connected, quasisimple algebraic group over an algebraically closed field of characteristic p > 0, and let V be a rational G-module such that dim V < p. According to a result of Jantzen, V is completely reducible, and H 1 (G, V) = 0. In this paper we show that H 2 (G, V) = 0 unless some composition factor of V is a nontrivial Frobenius twist of the adjoint representation of G.

The second cohomology of small irreducible modules for simple algebraic groups