Theory of Hopf Algebras Attached to Group Schemes Group schemes over a field Group subschemes of a group scheme Coalgebras over a field Hopf algebras over a field Structures of colocal bigebras Formal groups over a field Structures of formal groups as local rings Algebraic Hopf subalgebras Rational representations of group schemes Joins and intersections of group subschemes Adjoint representations of group schemes Normalizers of Hopf subalgebras and group subschemes Centralizers of Hopf subalgebras and group subschemes Commutators of Hopf subalgebras and group subschemes Lie algebras attached to group schemes Left invariant semi derivations of group varieties Algebraic schemes on which a group scheme operates Isogenies of group varieties Kernels of purely inseparable isogenies.