  Let O_K be a discrete valuation ring of mixed characteristics (0,p), with
residue field k. Using work of Sekiguchi and Suwa, we construct some finite
flat O_K-models of the group scheme \mu_{p^n,K} of p^n-th roots of unity, which
we call Kummer group schemes. We set carefully the general framework and
algebraic properties of this construction. When k is perfect and O_K is a
complete totally ramified extension of the ring of Witt vectors W(k), we
provide a parallel study of the Breuil-Kisin modules of finite flat models of
\mu_{p^n,K}, in such a way that the construction of Kummer groups and
Breuil-Kisin modules can be compared. We compute these objects for n < 4. This
leads us to conjecture that all finite flat models of \mu_{p^n,K} are Kummer
group schemes.
