This can be precisely phrased either by saying that a map f satisfies the equation fμ = μ(f × f), or by saying that f is a natural transformation of functors from schemes to groups (rather than just sets).
The multiplication, unit, and inverse maps of the group scheme are given by the comultiplication, counit, and antipode structures in the Hopf algebra.
For example, if 2 is invertible over the base, all group schemes of order 2 are constant, but over the 2-adic integers, μ2 is non-constant, because the special fiber isn't smooth.
There exist sequences of highly ramified 2-adic rings over which the number of isomorphism types of group schemes of order 2 grows arbitrarily large.
More detailed analysis of commutative finite flat group schemes over p-adic rings can be found in Raynaud's work on prolongations.
If we consider a family of elliptic curves, the p-torsion forms a finite flat group scheme over the parametrizing space, and the supersingular locus is where the fibers are connected.
This merging of connected components can be studied in fine detail by passing from a modular scheme to a rigid analytic space, where supersingular points are replaced by discs of positive radius.
Finite flat commutative group schemes over a perfect field k of positive characteristic p can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting.
Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze p-divisible groups.