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

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 $\lvert G/A\rvert\le25^{n^2}$.