# Arithmetic results on orbits of linear groups

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

Research output: Contribution to journalArticle

7 Citations (Scopus)

### 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 language English 2415-2467 Transactions of the American Mathematical Society 368 4 19 Aug 2015 https://doi.org/10.1090/tran/6373 Published - Apr 2016

### Fingerprint

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

TY - JOUR

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

Y1 - 2016/4

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.

U2 - 10.1090/tran/6373

DO - 10.1090/tran/6373

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VL - 368

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EP - 2467

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

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