For a reductive linear algebraic group G (over the complex numbers), all linear representations have the property that on an open dense subset of V, the stabilizers are all conjugate to each other. This is a result of Richardson and Luna. If H is an element of that conjugacy class, then any G-torsor (over a field extension of the complex numbers) is induced from an N_G(H)-torsor; in other words, we can reduce the structural group from G to the normalizer of H. This implies that the essential dimension of G is bounded above by that of N_G(H). I will discuss how this extends to more general base fields, in particular those of prime characteristic. The examples of G=F_4 and G=E_7 will be considered.