  We are interested in the actions of an algebraic group G over the
grassmannians of a finite dimensional K-vector space V (K algebraically closed)
deduced from an action of G over V. We prove that the dimension of the orbit
space of G(i,V) is smaller than that of G(j,V) if and only if the dimension of
G(i,V) is smaller than that of G(j,V). We then get similar results for flag
varieties. Different methods allow us to get results about the number of orbits
when the ground field is finite.
