OVERGROUPS OF SOME CLASSICAL LINEAR-GROUPS WITH APPLICATIONS TO LINEAR PRESERVER PROBLEMS

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