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
On some infinite dimensional linear groups
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
This is a survey indicating the newest results on infinite-dimensional linear groups.
Let F be a field, A – a vector space over F. The group GL(F, A) of all automorphisms of A and its distinct subgroups (the linear groups) are the oldest subjects of investigation in Group Theory. Naturally, the first step here was the investigation of the case, when A has finite dimension over F. Under this condition every element of GL(F, A) (a non-singular linear transformation) defines some non-singular n × n - matrix over F where n = dimF A. Thus, for the finite-dimensional case the theory of linear groups coincides with the theory of matrix groups. That is why the theory of finite-dimensional linear groups is one of the best-developed theories in algebra.
The case when dimF A is infinite is totally different. The study of this case always requires some additional essential restrictions. The situation here is similar to that one which appeared in the initial period of development of Infinite Group Theory. One of the very fruitful approaches is associated with the finiteness conditions. It seems very meaningful to apply it to the linear groups.
Take into consideration the following important aspect. Many problems related to groups with finiteness conditions require the study of modules over some group rings, frequently the group rings of the type FG where F is a field. Thus, the respectively finiteness conditions from groups could be transferred on the appropriate modules.
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- Groups St Andrews 2001 in Oxford , pp. 377 - 384Publisher: Cambridge University PressPrint publication year: 2003
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