Jordan property for non-linear algebraic groups and projective varieties

A century ago, Camille Jordan proved that the complex general linear group GLₙ(C) has the Jordan property: there is a Jordan constant Cₙ such that every finite subgroup H≤ GLₙ(C) has an abelian subgroup H₁ of index [H:H₁]≤ Cₙ. We show that every connected algebraic group G (which is not necessarily linear) has the Jordan property with the Jordan constant depending only on G, and that the full automorphism group Aut(X) of every projective variety X has the Jordan property.