Coprime subdegrees for primitive permutation groups and completely reducible linear groups

TL;DR

This paper investigates the conditions under which orbit sizes and subdegrees in finite linear and primitive permutation groups are coprime, revealing structural constraints and applications to field extensions.

Contribution

It establishes new results on coprime orbit lengths in linear and primitive permutation groups, linking group structure to orbit size properties.

Findings

Orbit length of a+b equals the product of individual orbit lengths for coprime orbits.

Irreducible linear groups have orbit lengths with non-trivial common factors.

Coprime non-identity subdegrees occur only in specific primitive group types.

Abstract

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.