Book contents
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Orbits in finite group actions
Published online by Cambridge University Press: 15 December 2009
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Summary
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.
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- Groups St Andrews 2001 in Oxford , pp. 306 - 331Publisher: Cambridge University PressPrint publication year: 2003
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