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Regular Subgroups of Primitive Permutation Groups
Martin W. Liebeck, Imperial College, London, England, Cheryl E. Praeger, University of Western Australia, Crawley, Australia, and Jan Saxl, University of Cambridge, England
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Memoirs of the American Mathematical Society
2009; 74 pp; softcover
Volume: 203
ISBN-10: 0-8218-4654-X
ISBN-13: 978-0-8218-4654-4
List Price: US$68 Individual Members: US$40.80
Institutional Members: US\$54.40
Order Code: MEMO/203/952

The authors address the classical problem of determining finite primitive permutation groups $$G$$ with a regular subgroup $$B$$. The main theorem solves the problem completely under the assumption that $$G$$ is almost simple. While there are many examples of regular subgroups of small degrees, the list is rather short (just four infinite families) if the degree is assumed to be large enough, for example at least 30!. Another result determines all primitive groups having a regular subgroup which is almost simple. This has an application to the theory of Cayley graphs of simple groups.

• Introduction
• Preliminaries
• Transitive and antiflag transitive linear groups
• Subgroups of classical groups transitive on subspaces
• Proof of Theorem 1.1: Linear groups
• Proof of Theorem 1.1: Unitary groups
• Proof of Theorem 1.1: Orthogonal groups in odd dimension
• Proof of Theorem 1.1: Orthogonal groups of minus type
• Proof of Theorem 1.1: Some special actions of symplectic and orthogonal groups
• Proof of Theorem 1.1: Remaining symplectic cases
• Proof of Theorem 1.1: Orthogonal groups of plus type
• Proof of Theorem 1.1: Exceptional groups of Lie type
• Proof of Theorem 1.1: Alternating groups
• Proof of Theorem 1.1: Sporadic groups
• Proof of Theorem 1.4 and Corollary 1.3
• The tables in Theorem 1.1
• References