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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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 PDF
Trans. Amer. Math. Soc. 356 (2004), 4857-4872 Request permission

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
  • 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