  We study magnetic properties of itinerant quantum magnetic particles
described by a generalized Hubbard model with large spin ($S>1/2$) which may be
realized in optical lattices of laser-cooled atom systems. In fermion systems
(half-integer spins), an extended form of Nagaoka ferromagnetism may be
realized. However, as novel aspects of the large spin cases, we found that the
condition on the lattice connectivity is more stringent than in the case of
$S=1/2$ particles, and that the system shows a peculiar degenerate structure of
the ground statein which the ferromagnetic state is included. In contrast, it
turns out that the ground state of itinerant bosonic systems (integer spins)
has a degenerate structure similar to that of fermion system with $S>1/2$
regardless of the shape, connectivity or filling of the lattice, and that the
state with the maximum total spin is always one of the ground states.
