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), 48574872
MSC (2000):
Primary 20B15, 20B30, 05C25
Published electronically:
January 13, 2004
MathSciNet review:
2084402
Fulltext PDF Free Access
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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 nontrivial 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 primepower order, then there exists a partition of such that is a TOD. This paper characterizes TODs such that is primitive and is cyclic of primepower order. An application is given to the classification of selfcomplementary vertextransitive 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/S0002994704033963
PII:
S 00029947(04)033963
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
