COPRIME SUBDEGREES FOR PRIMITIVE PERMUTATION GROUPS AND COMPLETELY REDUCIBLE LINEAR GROUPS

In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H ⊆ GL(V ) acting completely reducibly on a vector space V : if the H-orbits containing the vectors a and b have coprime lengths m and n, we prove that the H-orbit containing a + b has length mn. Such groups H are always reducible if n, m > 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.