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

TL;DR

This paper investigates finitely generated linear groups of finite rank with faithful irreducible primitive representations over characteristic zero fields, establishing that such groups have an infinite FC-center.

Contribution

It proves that infinite finitely generated linear groups of finite rank with faithful irreducible primitive representations must have an infinite FC-center.

Findings

Infinite finitely generated linear groups of finite rank with such representations have an infinite FC-center.

The study links primitive representations to the structure of the group's FC-center.

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.