  In this paper we answer a question of Gabriel Navarro about orbit sizes of a
finite linear group H acting completely reducibly on a vector space V: if the
orbits containing the vectors a and b have coprime lengths m and n, we prove
that the orbit containing a+b has length mn. Such groups H are always reducible
if n and m are greater than 1. In fact, if H is an irreducible linear group, we
show that, for every pair of non-zero vectors, their orbit lengths have a
non-trivial common factor.
  In the more general context of finite primitive permutation groups G, we show
that coprime non-identity subdegrees are possible if and only if G is of
O'Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a
finite primitive permutation group, a set of pairwise coprime subdegrees has
size at most 2.
  Finally, as an application of our results, we prove that a field has at most
2 finite extensions of pairwise coprime indices with the same normal closure.
