The concept of a synchronizing permutation group was introduced nearly 15 years ago as a possible way of approaching The Černý Conjecture. Such groups must be primitive. In an attempt to understand synchronizing groups, a whole hierarchy of properties for a permutation group has been developed, namely, 2-transitive groups, ℚI-groups, spreading, separating, synchronizing, almost synchronizing and primitive. Many surprising connections with other areas of mathematics such as finite geometry, graph theory, and design theory have arisen in the study of these properties. In this survey talk I will give an overview of the hierarchy and discuss what is known about which groups lie where.
This video was produced by Newcastle University, Australia, as part of the Symmetries in Newcastle seminar series.
