Orbital digraphs of infinite primitive permutation groups

Primitive permutation groups of bounded orbital diameter

We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive. A key result is that, in the almost simple case with socle of fixed Lie rank, apart from very specif...

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.

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 ...

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