Groups having a faithful irreducible representation

TL;DR

This paper investigates the conditions under which any group, finite or infinite, can possess a faithful irreducible representation over any field, aiming to generalize representation theory beyond finite groups.

Contribution

It provides necessary and sufficient criteria for groups to admit faithful irreducible representations over arbitrary fields, extending classical results to broader classes of groups.

Findings

Established criteria for finite groups.

Extended conditions to infinite groups.

Generalized representation theory results.

Abstract

We address the problem of finding necessary and sufficient conditions for an arbitrary group, not necessarily finite, to admit a faithful irreducible representation over an arbitrary field.