Coprime subdegrees for primitive permutation groups and completely reducible linear groups
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.
Cite
@article{arxiv.1109.6559,
title = {Coprime subdegrees for primitive permutation groups and completely reducible linear groups},
author = {Silvio Dolfi and Robert Guralnick and Cheryl Praeger and Pablo Spiga},
journal= {arXiv preprint arXiv:1109.6559},
year = {2011}
}
Comments
20 pages, 1 table