On orbital partitions and exceptionality of primitive permutation groups |
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| Authors: | R. M. Guralnick Cai Heng Li Cheryl E. Praeger J. Saxl |
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| Affiliation: | Department of Mathematics, University of Southern California, Los Angeles, California 90089 ; School of Mathematics and Statistics, The University of Western Australia, Crawley, Western Australia 6009, Australia ; School of Mathematics and Statistics, The University of Western Australia, Crawley, Western Australia 6009, Australia ; Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England |
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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 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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