Orbits in finite group actions

Abstract

Abstract We give an overview of results on the orbit structure of finite group actions with an emphasis on abstract linear groups. The main questions considered are the number of orbits, the number of orbit sizes and the existence of large orbits, particularly regular orbits. Applications of such results to some important problems in group and representation theory are also discussed, such as the k ( GV )–problem and the Taketa problem. Introduction Ever since the beginning of abstract group theory the study of group actions has played a fundamental role in its development. In finite group theory this is all too obvious in that the fundamental theorems of Sylow – without which finite group theory would not get beyond its beginnings – are based on the study of various group actions. Group actions capture the fact that every group can be represented as a permutation group, and permutation groups were the first groups to be considered when the abstract term of a group had not yet been coined. Naturally information on the orbits induced by a group action is vital to an understanding of the action which is why a wealth of such results is scattered throughout the literature. Rarely however are the orbits the main focus of the investigation, and most results on orbits are proved with applications to other questions in mind.