On orbital partitions and exceptionality of primitive permutation groups

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

Journal:
Trans. Amer. Math. Soc. **356** (2004), 4857-4872

MSC (2000):
Primary 20B15, 20B30, 05C25

DOI:
https://doi.org/10.1090/S0002-9947-04-03396-3

Published electronically:
January 13, 2004

MathSciNet review:
2084402

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

- Burton Fein, William M. Kantor, and Murray Schacher,
*Relative Brauer groups. II*, J. Reine Angew. Math.**328**(1981), 39–57. MR**636194**, DOI https://doi.org/10.1515/crll.1981.328.39 - Michael D. Fried, Robert Guralnick, and Jan Saxl,
*Schur covers and Carlitz’s conjecture*, Israel J. Math.**82**(1993), no. 1-3, 157–225. MR**1239049**, DOI https://doi.org/10.1007/BF02808112 - Daniel Gorenstein,
*Finite simple groups*, University Series in Mathematics, Plenum Publishing Corp., New York, 1982. An introduction to their classification. MR**698782** - Daniel Gorenstein and Richard Lyons,
*The local structure of finite groups of characteristic $2$ type*, Mem. Amer. Math. Soc.**42**(1983), no. 276, vii+731. MR**690900**, DOI https://doi.org/10.1090/memo/0276 - R. M. Guralnick, P. Müller, J. Saxl,
*The rational function analogue of a question of Schur and exceptionality of permutation representations*, Mem. Amer. Math. Soc. 162 (2003), no. 773. - Cai Heng Li,
*On self-complementary vertex-transitive graphs*, Comm. Algebra**25**(1997), no. 12, 3903–3908. MR**1481574**, DOI https://doi.org/10.1080/00927879708826094 - Cai Heng Li and Cheryl E. Praeger,
*Self-complementary vertex-transitive graphs need not be Cayley graphs*, Bull. London Math. Soc.**33**(2001), no. 6, 653–661. MR**1853775**, DOI https://doi.org/10.1112/S0024609301008505 - C. H. Li, C. E. Praeger, On partitioning the orbitals of a transitive permutation group,
*Trans. Amer. Math. Soc.***355**(2003), No. 2, 637-653. - Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl,
*Transitive subgroups of primitive permutation groups*, J. Algebra**234**(2000), no. 2, 291–361. Special issue in honor of Helmut Wielandt. MR**1800730**, DOI https://doi.org/10.1006/jabr.2000.8547 - Mikhail Muzychuk,
*On Sylow subgraphs of vertex-transitive self-complementary graphs*, Bull. London Math. Soc.**31**(1999), no. 5, 531–533. MR**1703877**, DOI https://doi.org/10.1112/S0024609399005925 - Horst Sachs,
*Über selbstkomplementäre Graphen*, Publ. Math. Debrecen**9**(1962), 270–288 (German). MR**151953**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
20B15,
20B30,
05C25

Retrieve articles in all journals with MSC (2000): 20B15, 20B30, 05C25

Additional Information

**R. M. Guralnick**

Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, California 90089

MR Author ID:
78455

Email:
guralnic@math.usc.edu

**Cai Heng Li**

Affiliation:
School of Mathematics and Statistics, The University of Western Australia, Crawley, Western Australia 6009, Australia

MR Author ID:
305568

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

MR Author ID:
141715

ORCID:
0000-0002-0881-7336

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

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