A complete classification is obtained for finite connected flat commutative group schemes over mixed characteristic complete discrete valuation rings. The group schemes are classified in terms of their Cartier modules. The equivalence of various definitions of the tangent space and the dimension for these group schemes is proved. This shows that the minimal dimension of a formal group law that contains a given connected group scheme S as a closed subgroup is equal to the minimal number of generators for the coordinate ring of S. The following reduction criteria for Abelian varieties are deduced. © 2007 American Mathematical Society.