From local to global similarity of matrix groups

Abstract

Let G and ℋ be groups of complex n×n matrices. We say that G is an ℋ-like group if every matrix in G is similar to a matrix from ℋ. For several groups ℋ we consider two questions:Is every ℋ-like group (simultaneously) similar to a subgroup of ℋ?Is ℋ the only ℋ-like group containing ℋ? Among other results we prove that the symmetric group Sn is the only Sn-like group containing Sn. © 2011 Elsevier Inc. All rights reserved.