Permutation Groups and Cartesian Decompositions

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.