TY - JOUR

T1 - Coprime subdegrees for primitive permutation groups and completely reducible linear groups

AU - Dolfi, S.

AU - Guralnick, R.M.

AU - Praeger, Cheryl

AU - Spiga, Pablo

PY - 2013

Y1 - 2013

N2 - 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. © 2013 Hebrew University Magnes Press.

AB - 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. © 2013 Hebrew University Magnes Press.

U2 - 10.1007/s11856-012-0163-4

DO - 10.1007/s11856-012-0163-4

M3 - Article

SN - 0021-2172

VL - 195

SP - 745

EP - 772

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 2

ER -