Abstract
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.
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The first author is supported by the MIUR project “Teoria dei gruppied applicazioni”.
The second author was partially supported by NSF grant DMS-1001962.
The third author is supported by the ARC Federation Fellowship Project FF0776186.
The fourth author is supported by the University of Western Australia as part of the Federation Fellowship project.
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Dolfi, S., Guralnick, R., Praeger, C.E. et al. Coprime subdegrees for primitive permutation groups and completely reducible linear groups. Isr. J. Math. 195, 745–772 (2013). https://doi.org/10.1007/s11856-012-0163-4
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DOI: https://doi.org/10.1007/s11856-012-0163-4