Groups acting on vector spaces with a large family of invariant subspaces

Abstract

Let A be a vector space over a field F, and let GL(F, A) be the group of all F-automorphisms of A. A subgroup G ≤ GL(F, A) is called a linear group (on A) and acts naturally on A as a group of (invertible) linear transformations. If B is a subspace of A, we say that B is G-invariant if B = Bg for every g ∈ G and set to denote the largest G-invariant subspace of A contained in B. In this article, we study the case when there exists a nonnegative integer b such that dim _F  (B/B _G ) ≤ b for every subspace B of A.

Keywords:

• vector space
• linear group
• invariant subspace

AMS Subject Classifications::

• 15A03
• 20F16
• 20F29

Acknowledgements

Part of this research was carried out while the second author visited the Department of Algebra of the National University of Dnepropetrovsk. He thanks everybody there for their warm hospitality during his visit. This research was supported by Proyecto MTM2010-19938-C03-03 of Dirección General de Investigación of MICINN (Spain). The second author was also supported by Caja de Ahorros de la Inmaculada (CAI) through the program CAI Programa Europa.