Closed H-orbits in G-varieties

Let G be a linear algebraic group, H a subgroup of G and X a G-variety. This paper explores the connection between G-orbits and H-orbits in X, concentrating in particular on the question of when we have the implications G.x closed in X implies H.x closed in X, and vice versa. Some of the general results found in this paper are then used in the special case where G and H are reductive and X is affine to extend two classical results of Luna from characteristic zero to arbitrary characteristic.