On the third cohomology of algebraic groups of rank two in positive characteristic

We evaluate the third cohomology of simple simply connected algebraic groups of rank 2 over an algebraically closed field of positive characteristic with coefficients in simple modules. It is assumed that the characteristic $p$ of the field is greater than $3$ for $\operatorname{SL}_3$, greater than $5$ for $\operatorname{Sp}_4$, and greater than $11$ for $G_2$. It follows from the main result that the dimensions of the cohomology spaces do not exceed the rank of the algebraic group in question. To prove the main results we study the properties of the first-quadrant Lyndon-Hochschild-Serre spectral sequence with respect to an infinitesimal subgroup, namely, the Frobenius kernel of the given algebraic group.
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