Primitive permutation groups and their section-regular partitions

On Orbital Partitions and Exceptionality of Primitive Permutation Groups

Let G and X be transitive permutation groups on a set Ω such that G is a normal subgroup of X. The overgroup X induces a natural action on the set Orbl(G,Ω) of non-trivial orbitals of G on Ω. In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples (G,X,Ω) where X fixes no elements of Orbl(G,Ω); such triples are called exceptional. In the study of ...

Partitions and Permutation Groups

We show that non-trivial extremely amenable topological groups are essentially the same thing as permutation models of the Boolean prime ideal theorem that do not satisfy the axiom of choice. Both are described in terms of partition properties of group actions.

Distinguishing Primitive Permutation Groups

Let G be a permutation group acting on a set V . A partition π of V is distinguishing if the only element of G that fixes each cell of π is the identity. The distinguishing number of G is the minimum number of cells in a distinguishing partition. We prove that if G is a primitive permutation group and |V | ≥ 336, its distinguishing number is two.

Factorizations of Primitive Permutation Groups

Suborbits in Infinite Primitive Permutation Groups

For every infinite cardinal κ, we construct a primitive permutation group which has a finite suborbit paired with a suborbit of size κ. This answers a question of Peter M. Neumann.