The theorem of Marggraff on primitive permutation groups which contain a cycle

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...

Doubly Transitive Permutation Groups Which Are Not Doubly Primitive

Hypothesis (A): G is a doubly transitive permutation group on a set Q. For 01 E Q, G, has a set Z = {B, , B, ,..., B,}, t > 2, which is a complete set of imprimitivity blocks on Q {a}. Let j Bi / = b > 1 for all i. Denote by H the kernel of G, on .Z and by Ki and K< the subgroups of G, fixing Bi setwise and pointwise respectively, 1 .< i < t. Let /3 E Bl . Here j Q j = 1 + ht. M. D. Atkinson ha...

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

On the tenacity of cycle permutation graph

A special class of cubic graphs are the cycle permutation graphs. A cycle permutation graph Pn(α) is defined by taking two vertex-disjoint cycles on n vertices and adding a matching between the vertices of the two cycles.In this paper we determine a good upper bound for tenacity of cycle permutation graphs.