Arithmetic results on orbits of linear groups

Michael Giudici, M.W. Liebeck, Cheryl Praeger, J. Saxl, P.H. Tiep

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    Abstract

    © 2015 American Mathematical Society. Let p be a prime and G a subgroup of GLd(p). We define G to be p-exceptional if it has order divisible by p, but all its orbits on vectors have size coprime to p. We obtain a classification of p-exceptional linear groups. This has consequences for a well-known conjecture in representation theory, and also for a longstanding question concerning 1/2 -transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by p.
    Original languageEnglish
    Pages (from-to)2415-2467
    JournalTransactions of the American Mathematical Society
    Volume368
    Issue number4
    Early online date19 Aug 2015
    DOIs
    Publication statusPublished - Apr 2016

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