A note about Jordan's bound on the size of finite linear groups

TL;DR

This paper revisits Jordan's classical result on finite linear groups, providing a streamlined proof that explicitly bounds the index of an abelian normal subgroup in terms of the dimension.

Contribution

It offers a simplified, self-contained proof of Frobenius's bound on finite linear groups, making the explicit bound more accessible.

Findings

Explicit bound $ vert G/A vert extless=25^{n^2}$ established.

Streamlined proof simplifies previous complex arguments.

Reinforces classical results with clearer presentation.

Abstract

In 1878 Camille Jordan showed that every finite subgroup $G\le\text{GL}_n(\mathbb C)$ has an abelian normal subgroup $A$ such that $\lvert G/A\rvert$ is bounded in terms of $n$, but he did not give an explicit bound. An explicit bound was obtained by Blichfeldt in a series of papers beginning in 1904, using representation-theoretic methods. In 1911 Bieberbach gave a geometric proof, which is quite different from the approaches of Jordan and Blichfeldt, together with an explicit bound. Frobenius simplified this proof in the same year, and the resulting argument is still the simplest known. We present a self-contained and streamlined variant of Frobenius's argument, yielding the bound $\lvert G/A\rvert\le25^{n^2}$.