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On orbital partitions and exceptionality of primitive permutation groups. (English) Zbl 1062.20001

Let \(G\) and \(X\) be transitive permutation groups on a set \(\Omega\) such that \(G\) is a normal subgroup of \(X\). The group \(X\) induces a natural action on the set \(\text{Orb}(G,\Omega)\) of non-trivial orbitals of \(G\) on \(\Omega\). If \(X\) fixes no element in \(\text{Orb}(G,\Omega)\), then the triple \((G,X,\Omega)\) is said to be exceptional. The exceptionality of transitive permutation groups arises naturally in the study of Galois groups of exceptional covers of curves [see M. D. Fried, R. Guralnick and J. Saxl, Isr. J. Math. 82, No. 1-3, 157-225 (1993; Zbl 0855.11063)]. Much information on exceptional triples may be found in the paper of R. M. Guralnick, P. Müller and J. Saxl [Mem. Am. Math. Soc. 773 (2003; Zbl 1082.12004)].
If \(\mathcal P\) is a partition of \(\text{Orb}(G,\Omega)\) such that \(X\) is transitive on \(\mathcal P\), then the quadruple \((G,X,\Omega,\mathcal P)\) is called a transitive orbital decomposition (TOD). TODs arise in a number of different areas [see C. H. Li and C. E. Praeger, Trans. Am. Math. Soc. 355, No. 2, 637-653 (2003; Zbl 1017.20001)]. For example, a TOD with \(|\mathcal P|=2\) exactly corresponds to a self-complementary vertex-transitive digraph. Such digraphs are studied, for example, by C. H. Li [Commun. Algebra 25, No. 12, 3903-3908 (1997; Zbl 0898.05030)], C. H. Li and C. E. Praeger [Bull. Lond. Math. Soc. 33, No. 6, 653-661 (2001; Zbl 1026.05058)], M. Muzychuk [Bull. Lond. Math. Soc. 31, No. 5, 531-533 (1999; Zbl 0928.05034)] and H. Sachs [Publ. Math. 9, 269-288 (1962; Zbl 0119.18904)].
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 \(\text{Orb}(G,\Omega)\) 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 digraphs.

MSC:

20B15 Primitive groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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