On orbital partitions and exceptionality of primitive permutation groups期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On orbital partitions and exceptionality of primitive permutation groups

Authors:	R M Guralnick Cai Heng Li Cheryl E Praeger J Saxl

Institution:	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

Abstract:	Let and be transitive permutation groups on a set $\Omega$ such that is a normal subgroup of . The overgroup induces a natural action on the set $\operatorname{Orbl}(G,\Omega)$ of non-trivial orbitals of on $\Omega$ . In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples $(G,X,\Omega)$ where 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 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 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 is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.

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