  We present an extension of Nagaoka's theorem in the SU($n$) generalization of
the infinite-$U$ Hubbard model. It is shown that, when there is exactly one
hole, the fully polarized states analogous to the ferromagnetic states in the
SU(2) Hubbard model are ground states. For a restricted class of models
satisfying the connectivity condition, these fully polarized states are the
unique ground states up to the trivial degeneracy due to the SU($n$) symmetry.
We also give examples of lattices in which the connectivity condition can be
verified explicitly. The examples include the triangular, kagome, and
hypercubic lattices in $d (\ge 2)$ dimensions, among which the cases of $d=2$
and 3 are experimentally realizable in ultracold atomic gases loaded into
optical lattices.
