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In this paper we answer a question of Gabriel Navarro about orbit sizes of afinite linear group H acting completely reducibly on a vector space V: if theorbits containing the vectors a and b have coprime lengths m and n, we provethat the orbit containing a+b has length mn. Such groups H are always reducibleif n and m are greater than 1. In fact, if H is an irreducible linear group, weshow that, for every pair of non-zero vectors, their orbit lengths have anon-trivial common factor.  In the more general context of finite primitive permutation groups G, we showthat coprime non-identity subdegrees are possible if and only if G is ofO'Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for afinite primitive permutation group, a set of pairwise coprime subdegrees hassize at most 2.  Finally, as an application of our results, we prove that a field has at most2 finite extensions of pairwise coprime indices with the same normal closure.

Coprime subdegrees for primitive permutation groups and completely reducible linear groups