Classifying finite monomial linear groups of prime degree in characteristic zero

TL;DR

This paper provides a comprehensive classification of finite solvable irreducible monomial subgroups of GL(p,C) for prime p, including structural insights and a nearly complete classification of non-solvable cases, with results available in Magma.

Contribution

It offers the first explicit classification of finite solvable irreducible monomial groups of prime degree in characteristic zero and discusses obstacles in classifying non-solvable cases.

Findings

Complete list of conjugacy class representatives for solvable groups.

Structural analysis of non-solvable monomial subgroups.

Classification of all finite irreducible subgroups for p ≤ 3.

Abstract

Let $p$ be a prime and let $\mathbb{C}$ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ up to conjugacy. That is, we give a complete and irredundant list of $\mathrm{GL}(p,\mathbb{C})$-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For $p\leq 3$, we classify all finite irreducible subgroups of $\mathrm{GL}(p,\mathbb{C})$. Our classifications are available publicly in Magma.