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Permutation Groups and Cartesian Decompositions

Cheryl E. Praeger and Csaba Schneider
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Lecture Note Series 449
[Reviewed by
Miklós Bóna
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This is a thorough reference book that consists of three parts. The first part introduces permutation groups. A non-expert reader with graduate student’s knowledge of permutation groups will probably find the second half of even this introductory part new.

The second, main part is about innately transitive groups and their cartesian decompositions. A permutation group is called quasiprimitive if all its nontrivial normal subgroups are transitive, and it is called innately transitive if it has a transitive minimal normal subgroup. The second crucial notion is that of cartesian decompositions. A cartesian decompostion \(C\) of a set \(\Omega\) is a set \(\{\Gamma_1,\Gamma_2,\cdots ,\Gamma_k\}\) of set partitions of \(\Omega\), in which each \(\Gamma_i\) has at least two blocks, and for which the condition \[|\gamma_1\cap \gamma_2 \cap \cdots \cap \gamma_k | =1\] holds for all choices of the \(\gamma_i\), where \(\gamma_i\) is a block of \(\Gamma_i\). A central question that the authors consider is whether for a given innately transitive group \(G\) acting on a set \(\Omega\), there is a nontrivial cartesian decomposition of \(\Omega\) that \(G\) leaves invariant. (The trivial carteseian decomposition is the one that consists of only one partition, the all-singleton partition.)

The book ends with a short third part consisting of applications, to other parts of the theory of permutation groups, and to graph theory. In summary, the book is an impressive collection of theorems and their proofs. There are very few examples and no exercises. For this reason, teaching a class from the book could be very difficult, and it seems that most readers will use the book as reference material.

Miklós Bóna is Professor of Mathematics at the University of Florida.

1. Introduction
Part I. Permutation Groups – Fundamentals:
2. Group actions and permutation groups
3. Minimal normal subgroups of transitive permutation groups
4. Finite direct products of groups
5. Wreath products
6. Twisted wreath products
7. O'Nan–Scott theory and the maximal subgroups of finite alternating and symmetric groups
Part II. Innately Transitive Groups – Factorisations and Cartesian Decompositions:
8. Cartesian factorisations
9. Transitive cartesian decompositions for innately transitive groups
10. Intransitive cartesian decompositions
Part III. Cartesian Decompositions – Applications:
11. Applications in permutation group theory
12. Applications to graph theory
Appendix. Factorisations of simple and characteristically simple groups