Grupos sincronizadores e o teorema de O'nan-Scott

Abstract

The O'Nan-Scott Theorem classifies the finite primitive permutation groups by dividing them into eight classes, according to the structure of its minimal normal subgroups. An important result in this classification is that every finite primitive permutation group admits at most two minimal normal subgroups. The synchronization property emerged from finite automata and transformation semigroup theory. Synchronizing permutation groups were introduced by Arnold [1] and Steinberg [2] to study the Cerný Conjecture from another perspective. Peter Neumann [12] studied the synchronizing groups using graph theory. He introduced a graph theoretic characterization of non-synchronizing groups in terms of undirected orbital graphs, and proved that a primitive group that preserves a \power structure" can not be synchronizing. Peter Cameron [5] also contributed in this area, and defined the so called basic groups. It is not dificult to show that synchronizing groups are primitive. This motivates us to inquire in which of the classes in the O'Nan-Scott Theorem we can find synchronizing groups. In this work, we use the concept of cartesian decompositions, introduced by Kovács [9] and reformulated by Baddeley, Praeger and Schneider [3], to show that in four, among the eight classes of the O'Nan-Scott Theorem, there are no synchronizing groups. This tells us that the synchronizing groups can be found only in the remaining four classes, but we will see that even in these classes there are non-synchronizing groups.