Coprime subdegrees for primitive permutation groups and completely reducible linear groups

Details Coprime subdegrees for primitive permutation groups and completely reducible linear groups Coprime subdegrees for primitive permutation groups and completely reducible linear groups

2011-09-29

arxiv.org/abs/1109.6559v2

Mathematics

Group Theory

20 pages, 1 table

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Coprime subdegrees for primitive permutation groups and completely reducible linear groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Coprime subdegrees for primitive permutation groups and completely reducible linear groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Coprime subdegrees for primitive permutation groups and completely reducible linear groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-152330