  It is argued that ultracold quantum degenerate gas of ytterbium $^{173}$Yb
atoms having nuclear spin $I = 5/2$ exhibits an enlarged SU$(6)$ symmetry.
Within the Landau Fermi liquid theory, stability criteria against Fermi liquid
(Pomeranchuk) instabilities in the spin channel are considered. Focusing on the
SU$(n > 2)$ generalizations of ferromagnetism, it is shown within mean-field
theory that the transition from the paramagnet to the itinerant ferromagnet is
generically first order. On symmetry grounds, general SU$(n)$ itinerant
ferromagnetic ground states and their topological excitations are also
discussed. These SU$(n > 2)$ ferromagnets can become stable by increasing the
scattering length using optical methods or in an optical lattice. However, in
an optical lattice at current experimental temperatures, Mott states with
different filling are expected to coexist in the same trap, as obtained from a
calculation based on the SU$(6)$ Hubbard model.
