Overgroups of some classical linear groups with applications to linear preserver problems

Abstract

The study of linear operators on a matrix space that leave invariant certain functions, subsets, or relations is commonly referred to as the study of linear preserver problems, and has attracted the attention of many mathematicians in the last few decades. Dynkin in his classic paper studied maximal subgroups of the classical groups and showed how his results may be used to study preserver problems. The purpose of this paper is to further exploit this very powerful approach. The subgroups G of the general linear group that we deal with are not necessarily maximal. For the applications that we have in mind we obtained a description of all possible overgroups of G. The results are then applied to various linear preserver problems. Shorter alternative proofs for various existing results are given, and some open questions are answered.