Marina Anagnostopoulou-Merkouri

Permutation groups, partition lattices and block structures.

Let G<= Sym(\Omega) be a finite transitive permutation group. We say that G is primitive if it preserves no nontrivial partition of \Omega, and imprimitive otherwise. Primitive groups are essential in the study of permutation group theory, as they are in some sense the `building blocks’ of all permutation groups and they have been a central object of study since the very beginning of group theory. Not as much is known on the other hand about imprimitive groups and there is no sensible way to describe their structure similar to the O’Nan-Scott theorem for primitive groups. However, those groups are very interesting to study from a combinatorial angle. In particular, it can be very fruitful to explore the lattice of their invariant partitions. In this talk we will present some recent work on imprimitive groups whose lattices of partitions form special combinatorial structures called orthogonal and poset block structures, which are widely used by statisticians in the design of experiments. This is joint work with Rosemary Bailey and Peter Cameron.