The Research About A Type Of Affine Primitive Non-synchronizing Groups

In this thesis, we will study the synchronizing groups and some relevant prob-lems concerning it. The synchronization property emerged from finite automata in the semigroup theory, more precisely, from the famous Cerny Conjecture. In the recent ten years the study of synchronizing groups has been the forefront research in the permutation group theory. Since the synchronizing groups are primitive groups, and primitive groups play an essential role in the permutation group theory, the study of synchronizing groups is very important in the permutation group theory. It is known that synchronizing groups are the so called basic groups, and by basic groups we mean that of the affine type, the almost simple type or the simple diago-nal type primitive groups. So synchronizing groups are one of the affine type, almost simple type or the simple diagonal type. Therefore, we will come up with a basic topic:what kind of primitive groups are non-synchronization? In this thesis we will study the generalized Paley graphs and discuss the condition of the equality between the clique number and chromatic number of the generalized Paley graphs. Then, we will prove that the generalized Paley graphs is isomorphic to the orbital graphs of a certain affine groups, and discuss the synchronization property of a certain affine groups. Furthermore, we will use this property to construct a type of primitive non-synchronizing groups. As an application of synchronization property, we will give a simple proof that one-dimensional affine groups are synchronization. Then we will discuss some properties of the special primitive non-synchronizing groups so that we will have a better understanding about the affine primitive non-synchronizing groups.