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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On orbital partitions and exceptionality of primitive permutation groups


Authors: R. M. Guralnick, Cai Heng Li, Cheryl E. Praeger and J. Saxl
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4857-4872
MSC (2000): Primary 20B15, 20B30, 05C25
Published electronically: January 13, 2004
MathSciNet review: 2084402
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Abstract: Let $G$ and $X$ be transitive permutation groups on a set $\Omega$ such that $G$ is a normal subgroup of $X$. The overgroup $X$ induces a natural action on the set $\operatorname{Orbl}(G,\Omega)$ of non-trivial orbitals of $G$ on $\Omega$. In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples $(G,X,\Omega)$ where $X$fixes no elements of $\operatorname{Orbl}(G,\Omega)$; such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples $(G,X,\Omega,\mathcal{P})$ where $\mathcal{P}$ is a partition of $\operatorname{Orbl}(G,\Omega)$ such that $X$ is transitive on $\mathcal{P}$; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple $(G,X,\Omega)$ in a TOD $(G,X,\Omega,\mathcal{P})$is exceptional; conversely if an exceptional triple $(G,X,\Omega)$ is such that $X/G$ is cyclic of prime-power order, then there exists a partition $\mathcal{P}$ of $\operatorname{Orbl}(G,\Omega)$ such that $(G,X,\Omega,\mathcal{P})$ is a TOD. This paper characterizes TODs $(G,X,\Omega,\mathcal{P})$ such that $X^\Omega$ is primitive and $X/G$ is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.


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Additional Information

R. M. Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
Email: guralnic@math.usc.edu

Cai Heng Li
Affiliation: School of Mathematics and Statistics, The University of Western Australia, Crawley, Western Australia 6009, Australia
Email: li@maths.uwa.edu.au

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

J. Saxl
Affiliation: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England
Email: saxl@dpmms.cam.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03396-3
PII: S 0002-9947(04)03396-3
Received by editor(s): October 5, 2002
Received by editor(s) in revised form: April 15, 2003
Published electronically: January 13, 2004
Additional Notes: This paper is part of a project funded by the Australian Research Council. The first author acknowledges support from NSF grant DMS 0140578, and the first and fourth authors acknowledge support by an EPSRC grant.
Article copyright: © Copyright 2004 American Mathematical Society