Finite subgroups of algebraic groups

Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of $\operatorname{GL}_n$ over a field of any characteristic $p$ possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic $p$, a commutative group of order prime to $p$, and a $p$-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.