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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    Linear Group
    Divisible
    Orbits
    Orbit
    Coprime
    Representation Theory
    Subgroup

    Cite this

    Giudici, Michael ; Liebeck, M.W. ; Praeger, Cheryl ; Saxl, J. ; Tiep, P.H. / Arithmetic results on orbits of linear groups. In: Transactions of the American Mathematical Society. 2016 ; Vol. 368, No. 4. pp. 2415-2467.
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    Arithmetic results on orbits of linear groups. / Giudici, Michael; Liebeck, M.W.; Praeger, Cheryl; Saxl, J.; Tiep, P.H.

    In: Transactions of the American Mathematical Society, Vol. 368, No. 4, 04.2016, p. 2415-2467.

    Research output: Contribution to journalArticle

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    T1 - Arithmetic results on orbits of linear groups

    AU - Giudici, Michael

    AU - Liebeck, M.W.

    AU - Praeger, Cheryl

    AU - Saxl, J.

    AU - Tiep, P.H.

    PY - 2016/4

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    N2 - © 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.

    AB - © 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.

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    DO - 10.1090/tran/6373

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    JO - Transactions of the American Mathematical Society

    JF - Transactions of the American Mathematical Society

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