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Oligomorphic permutation groups. (English) Zbl 0813.20002

London Mathematical Society Lecture Note Series. 152. Cambridge: Cambridge University Press. viii, 160 p. (1990).
This book was written mainly in 1988, immediately after the Durham Symposium on ‘Model Theory and Groups’ in that year. It stays close to the themes of that conference, but is not a Proceedings but a highly individual account of a beautiful area of mathematics, largely the author’s creation.
An oligomorphic permutation group is a permutation group \(G\) on a countably infinite set \(\Omega\) such that, for any positive integer \(n\), \(G\) has finitely many orbits in its induced action on \(\Omega^ n\). Equivalently (by the theorem of Engeler, Ryll-Nardzewski and Svenonius), if we endow \(\text{Sym}(\Omega)\) with a natural topology then \(G\) is oligomorphic if it is a dense subgroup of the automorphism group of some \(\omega\)-categorical structure on \(\Omega\). This accounts for the mixture of permutation group theory and model theory in the book.
The first chapter gives a rapid and readable account of some topics and tools needed later: a little permutation group theory and model theory; measure and Baire category; Ramsey’s Theorem. The second chapter then fuses the language of permutation group theory with that of model theory via the Ryll-Nardzewski Theorem and also Fraïsse’s Theorem on homogeneous structures. These are discussed and mostly proved. Some examples are introduced, and the topology mentioned above is described. Taken together, these two chapters constitute an excellent apéritif for model theory for a graduate student.
The core of the book is in Chapters 3 and 4. If \((G,\Omega)\) is an oligomorphic permutation group, then there are associated two natural sequences of positive integers: \(f_ n\) denotes the number of orbits of \(G\) on the set of unordered \(n\)-subsets of \(\Omega\), and \(F_ n\) the number of orbits on ordered \(n\)-sets. Both are non-decreasing, and there are many intriguing combinatorial questions involving their growth rates, local behaviour (when can one have \(f_ n = f_{n + 1} > 1\)?), the combinatorial sequences realised as \(f_ n(G)\) for some \(G\), the relationship between the \(f_ n\) and the \(F_ n\), and the relationships between these numbers for a wreath product and its constituent pieces. These topics were mostly developed by the author, and he describes the area in Chapter 3.
Chapter 4 concerns subgroups of oligomorphic groups. It begins with some results on the existence of free subgroups of closed oligomorphic groups. Powerful results are then proved by techniques of measure and Baire category. For example, as an infinitary counterpoint to Schur’s results on finite \(B\)-groups, for a very rich class \(\mathcal C\) of countable groups, every member of \(\mathcal C\) is isomorphic to a regular subgroup of the automorphism group of the random graph. This result (due to Cameron and K. W. Johnson) and some relatives are surveyed. There is also a brief discussion of the small index property and the normal subgroup structure of a closed oligomorphic group.
The final chapter consists of miscellaneous topics, including: Jordan groups; the question of when the ‘forth’ part of the back-and-forth argument suffices to build an automorphism; \(\omega\)-categorical, \(\omega\)-stable structures.
The book is written in a relaxed style, never straying from the elegant, and is a pleasure to read. Many of the topics are surveyed lightly, and the reader should expect to find sketches but not full proofs of the harder results mentioned. It is rich in ideas, and excellent, for example, as a source of problems for a beginning graduate student.

MSC:

20B07 General theory for infinite permutation groups
03C60 Model-theoretic algebra
20-02 Research exposition (monographs, survey articles) pertaining to group theory
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