## On orbital partitions and exceptionality of primitive permutation groups

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- by R. M. Guralnick, Cai Heng Li, Cheryl E. Praeger and J. Saxl
- Trans. Amer. Math. Soc.
**356**(2004), 4857-4872 - DOI: https://doi.org/10.1090/S0002-9947-04-03396-3
- Published electronically: January 13, 2004
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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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## Bibliographic Information

**R. M. Guralnick**- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
- MR Author ID: 78455
- Email: guralnic@math.usc.edu
**Cai Heng Li**- Affiliation: School of Mathematics and Statistics, The University of Western Australia, Crawley, Western Australia 6009, Australia
- MR Author ID: 305568
- 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
- MR Author ID: 141715
- ORCID: 0000-0002-0881-7336
- 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
- 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.
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**356**(2004), 4857-4872 - MSC (2000): Primary 20B15, 20B30, 05C25
- DOI: https://doi.org/10.1090/S0002-9947-04-03396-3
- MathSciNet review: 2084402