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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On partitioning the orbitals of a transitive permutation group


Authors: Cai Heng Li and Cheryl E. Praeger
Journal: Trans. Amer. Math. Soc. 355 (2003), 637-653
MSC (2000): Primary 20B15, 20B30, 05C25
Published electronically: September 19, 2002
MathSciNet review: 1932718
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Abstract: Let $G$ be a permutation group on a set $\Omega$ with a transitive normal subgroup $M$. Then $G$ acts on the set $\mathrm{Orbl}(M,\Omega)$ of nontrivial $M$-orbitals in the natural way, and here we are interested in the case where $\mathrm{Orbl}(M,\Omega)$ has a partition $\mathcal P$ such that $G$ acts transitively on $\mathcal P$. The problem of characterising such tuples $(M,G,\Omega,\mathcal P)$, called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where $\vert\mathcal P\vert$ is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where $\vert\mathcal P\vert=2$ exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the $G$-actions on $\Omega$ and on $\mathcal P$, and gives some construction methods for TODs.


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Additional Information

Cai Heng Li
Affiliation: Department of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia
Email: li@maths.uwa.edu.au

Cheryl E. Praeger
Affiliation: Department of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia
Email: praeger@maths.uwa.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03110-0
PII: S 0002-9947(02)03110-0
Received by editor(s): October 23, 2001
Published electronically: September 19, 2002
Additional Notes: This work forms a part of an Australian Research Council grant project
Article copyright: © Copyright 2002 American Mathematical Society