On irreducibility of induced modules and an adaptation of the Wigner--Mackey method of little groups

TL;DR

This paper establishes conditions for the irreducibility of induced modules and constructs irreducible representations for semidirect product groups over fields, extending classical methods without requiring algebraic closure.

Contribution

It provides new sufficiency conditions for irreducibility and adapts the Wigner--Mackey method for a broader class of groups and fields.

Findings

Derived new criteria for irreducibility of induced modules

Constructed explicit irreducible representations for semidirect product groups

Extended classical representation theory methods to non-algebraically closed fields

Abstract

This paper deals with sufficiency conditions for irreducibility of certain induced modules. We also construct irreducible representations for a group $G$ over a field ${\mathbb K}$ where the group $G$ is a semidirect product of a normal abelian subgroup $N$ and a subgroup $H$. The main results are proved with the assumption that ${\rm char} {\mathbb K}$ does not divide $|G|$ but there is no assumption made of ${\mathbb K}$ being algebraically closed.