On the finite irreducible subgroups of $\mathrm{GL}_n(\mathbb{C})$ and $\mathrm{PGL}_n(\mathbb{C})$

Abstract

The classification of the finite subgroups of $\mathrm{GL}_n(\mathbb{C})$ and $\mathrm{PGL}_n(\mathbb{C})$ is a classical problem in the field of finite group theory, dating back to the late 19th century with authors like Klein, Jordan, Blichfeldt, etc. Throughout its long history, many results concerning the classification have been scattered in the mathematical literature. In this survey, we explore many of the most relevant results related to the classification, its structure, and the lists of groups. We mostly focus on the irreducible groups. In particular, classification statements are provided for primitive and imprimitive groups over prime dimension, and quasi-primitive groups of small composite dimension. We also provide tables with a detailed list of all (quasi)-primitive finite groups of dimension $n<8$. Finally, we provide a computer program implementing many of the known results specific to the finite quasisimple irreducible projective groups, to improve and preserve the accessibility to these results and further their classification. See \S 4.2 and [37]. We aim this survey to the non-specialist, so the provided classification results may easily accessible to mathematicians working outside of group theory.