Extensions of a theorem of Jordan on primitive permutation groups

On the Orders of Primitive Permutation Groups

The problem of bounding the order of a permutation group G in terms of its degree n was one of the central problems of 19th century group theory (see [4]). It is closely related to the 1860 Grand Prix problem of the Paris Academy, but its history goes in fact much further back (see e.g. [3], [1] and [10]). The heart of the problem is of course the case where G is a primitive group. The best res...

Factorizations of Primitive Permutation Groups

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.

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...

Closures of Finite Primitive Permutation Groups

Let G be a primitive permutation group on a finite set ft, and, for k ^ 2, let G be the Ar-closure of G, that is, the largest subgroup of Sym (ft) preserving all the G-invariant ^-relations on ft. Suppose that G<H^ G and G and H have different socles. It is shown that k ^ 5 and the groups G and H are classified explicitly.