This book applies model theoretic methods to the study of certain finite
permutation groups, the automorphism groups of structures for a fixed
finite language with a bounded number of orbits on 4-tuples. Primitive
permutation groups of this type have been classified by Kantor, Liebeck,
and Macpherson, using the classification of the finite simple groups.
Building on this work, Gregory Cherlin and Ehud Hrushovski here treat
the general case by developing analogs of the model theoretic methods of
geometric stability theory. The work lies at the juncture of permutation
group theory, model theory, classical geometries, and combinatorics.
The principal results are finite theorems, an associated analysis of
computational issues, and an "intrinsic" characterization of the
permutation groups (or finite structures) under consideration. The main
finiteness theorem shows that the structures under consideration fall
naturally into finitely many families, with each family parametrized by
finitely many numerical invariants (dimensions of associated
coordinating geometries).
The authors provide a case study in the extension of methods of stable
model theory to a nonstable context, related to work on Shelah's "simple
theories." They also generalize Lachlan's results on stable homogeneous
structures for finite relational languages, solving problems of
effectivity left open by that case. Their methods involve the analysis
of groups interpretable in these structures, an analog of Zilber's
envelopes, and the combinatorics of the underlying geometries. Taking
geometric stability theory into new territory, this book is for
mathematicians interested in model theory and group theory.
