On the Primitive Irreducible Representations of Finitely Generated Linear Groups of Finite Rank

Abstract: In the paper we study finitely generated linear groups of finite rank which have faithful irreducible primitive representations over a field of characteristic zero. We prove that if an infinite finitely generated linear group $G$ of finite rank has a faithful irreducible primitive representation over a field of characteristic zero then the $FC$-center $\Delta (G)$ of $G$ is infinite.