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

DOI:
https://doi.org/10.1090/S0002-9947-04-03396-3

Published electronically:
January 13, 2004

MathSciNet review:
2084402

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Abstract | References | Similar Articles | Additional Information

Abstract: Let and be transitive permutation groups on a set such that is a normal subgroup of . The overgroup induces a natural action on the set of non-trivial orbitals of on . In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples where fixes no elements of ; such triples are called *exceptional*. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples where is a partition of such that is transitive on ; such a quadruple is called a *TOD* (transitive orbital decomposition). It follows easily that the triple in a TOD is exceptional; conversely if an exceptional triple is such that is cyclic of prime-power order, then there exists a partition of such that is a TOD. This paper characterizes TODs such that is primitive and 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:
https://doi.org/10.1090/S0002-9947-04-03396-3

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