Finite subgroups of algebraic groups

Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of GL n GL_n over a field of any characteristic p p possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic p p , a commutative group of order prime to p p , and a p 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.