**Memoirs of the American Mathematical Society**

2009;
74 pp;
Softcover

MSC: Primary 20; 05;

Print ISBN: 978-0-8218-4654-4

Product Code: MEMO/203/952

List Price: $68.00

AMS Member Price: $40.80

MAA Member Price: $61.20

**Electronic ISBN: 978-1-4704-0566-3
Product Code: MEMO/203/952.E**

List Price: $68.00

AMS Member Price: $40.80

MAA Member Price: $61.20

# Regular Subgroups of Primitive Permutation Groups

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*Martin W. Liebeck; Cheryl E. Praeger; Jan Saxl*

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.

#### Table of Contents

# Table of Contents

## Regular Subgroups of Primitive Permutation Groups

- Chapter 1. Introduction 18 free
- Chapter 2. Preliminaries 714 free
- Chapter 3. Transitive and antiflag transitive linear groups 1118
- Chapter 4. Subgroups of classical groups transitive on subspaces 1522
- Chapter 5. Proof of Theorem 1.1: Linear groups 2330
- Chapter 6. Proof of Theorem 1.1: Unitary groups 2734
- Chapter 7. Proof of Theorem 1.1: Orthogonal groups in odd dimension 3138
- Chapter 8. Proof of Theorem 1.1: Orthogonal groups of minus type 3542
- Chapter 9. Proof of Theorem 1.1: Some special actions of symplectic and orthogonal groups 3744
- Chapter 10. Proof of Theorem 1.1: Remaining symplectic cases 4552
- Chapter 11. Proof of Theorem 1.1: Orthogonal groups of plus type 5158
- Chapter 12. Proof of Theorem 1.1: Exceptional groups of Lie type 5562
- Chapter 13. Proof of Theorem 1.1: Alternating groups 5764
- Chapter 14. Proof of Theorem 1.1: Sporadic groups 6168
- Chapter 15. Proof of Theorem 1.4 and Corollary 1.3 6572
- Chapter 16. The tables in Theorem 1.1 6976
- References 7380