Coprime subdegrees for primitive permutation groups and completely reducible linear groups

S. Dolfi, R.M. Guralnick, Cheryl Praeger, Pablo Spiga

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    7 Citations (Scopus)
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    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. © 2013 Hebrew University Magnes Press.
    Original languageEnglish
    Pages (from-to)745-772
    JournalIsrael Journal of Mathematics
    Volume195
    Issue number2
    DOIs
    Publication statusPublished - 2013

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