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

• Published in: Linear & multilinear algebra Vol. 60; no. 4; pp. 487 - 498
• Main Authors: Kurdachenko, L.A., Muñoz-Escolano, J.M., Otal, J.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis Group 01-04-2012; Taylor & Francis Ltd
• Subjects: Algebra; Group theory; Integers; invariant subspace; Invariants; linear group; Linear transformations; Subgroups; Subspaces; Vector space; Vector spaces
• ISSN: 0308-1087; 1563-5139
• DOI: 10.1080/03081087.2011.608667

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.