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

Peter Müller

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

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

Abstract

In 1878 Camille Jordan showed that every finite subgroup has an abelian normal subgroup such that is bounded in terms of , 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 .
Paper Structure (4 sections, 3 theorems, 8 equations)

This paper contains 4 sections, 3 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Normed vector spaces and linear isometries
  3. Commuting elements
  4. The ball packing argument

Key Result

Theorem 1.1

Let $G\le\mathop{\mathrm{GL}}\nolimits_n(\mathbb C)$ be a finite group. Then $G$ has an abelian normal subgroup $A$ such that $\lvert G/A\rvert\le f(n)$, where $f(n)$ depends only on $n$.

Theorems & Definitions (6)

  • Theorem 1.1: Jordan 1878
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Remark 4.1