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The classification of $ \frac{3}{2}$-transitive permutation groups and $ \frac{1}{2}$-transitive linear groups


Authors: Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl
Journal: Proc. Amer. Math. Soc. 147 (2019), 5023-5037
MSC (2010): Primary 20B05, 20B15, 20B20, 20C15
DOI: https://doi.org/10.1090/proc/13243
Published electronically: September 23, 2019
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Abstract: A linear group $ G\le GL(V)$, where $ V$ is a finite vector space, is called $ \frac {1}{2}$-transitive if all the $ G$-orbits on the set of nonzero vectors have the same size. We complete the classification of all the $ \frac {1}{2}$-transitive linear groups. As a consequence we complete the determination of the finite $ \frac {3}{2}$-transitive permutation groups - the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the $ (k+\frac {1}{2})$-transitive groups for integers $ k\ge 2$.


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Additional Information

Martin W. Liebeck
Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom
Email: m.liebeck@imperial.ac.uk

Cheryl E. Praeger
Affiliation: School of Mathematics and Statistics, University of Western Australia, Western Australia 6009
Email: praeger@maths.uwa.edu.au

Jan Saxl
Affiliation: DPMMS, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

DOI: https://doi.org/10.1090/proc/13243
Received by editor(s): May 14, 2015
Published electronically: September 23, 2019
Additional Notes: The second author acknowledges the support of Australian Research Council Discovery Project Grant DP140100416.
Dedicated: Dedicated to our friend and teacher Peter Neumann on the occasion of his 75th birthday
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2019 American Mathematical Society