Jordan property for non-linear algebraic groups and projective varieties

A century ago, Camille Jordan proved that the complex general linear group $G{L}_n(C)$ has the Jordan property: there is a Jordan constant $C_n$ such that every finite subgroup $H\le G{L}_n(C)$ has an abelian subgroup $H_1$ of index $[H:H_1]\le C_n$. We show that every connected algebraic group $G$ (which is not necessarily linear) has the Jordan property with the Jordan constant depending only on $\dim G$, and that the full automorphism group $Aut(X)$ of every projective variety $X$ has the Jordan property