On the irreducible representations of soluble groups of finite rank

TL;DR

This paper investigates conditions under which soluble groups of finite rank have faithful irreducible representations over various fields, emphasizing the role of the group's socle and the complexities introduced by locally finite fields.

Contribution

It provides necessary and sufficient conditions for the existence of faithful irreducible representations of soluble groups of finite rank, highlighting the importance of the group's socle structure.

Findings

Existence depends on the construction of the group's socle.

Conditions vary significantly when the field is locally finite.

The paper clarifies the relationship between group structure and representation theory.

Abstract

We obtained some sufficient and necessary conditions of existence of faithful irreducible representations of a soluble group $G$ of finite rank over a field $k$. It was shown that the existence of such representations strongly depends on construction of the socle of the group $G$. The situation is especially complicated in the case where the field $k$ is locally finite.