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Finite p-Irregular Subgroups of PGL(2,k)

Xander Faber

Abstract

In the late 19th century, Klein inaugurated a program for describing the finite subgroups of $PGL_2(k)$ by treating the case in which the field $k$ is the complex numbers. Gierster and Moore extended Klein's arguments to deal with finite fields. In the past century, additional contributions to this problem were made by Serre, Suzuki, and Beauville, among others. We complete this program by giving a classification of the finite subgroups of $PGL_2(k)$ with order divisible by $p$, up to conjugation, for an arbitrary field $k$ of positive characteristic $p$.

Finite p-Irregular Subgroups of PGL(2,k)

Abstract

In the late 19th century, Klein inaugurated a program for describing the finite subgroups of by treating the case in which the field is the complex numbers. Gierster and Moore extended Klein's arguments to deal with finite fields. In the past century, additional contributions to this problem were made by Serre, Suzuki, and Beauville, among others. We complete this program by giving a classification of the finite subgroups of with order divisible by , up to conjugation, for an arbitrary field of positive characteristic .

Paper Structure

This paper contains 25 sections, 38 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. The Classification
  3. Fixed points
  4. Special subgroups
  5. Cyclic subgroups
  6. -subgroups
  7. Subgroups stabilizing a pair of points
  8. Subgroups fixing a unique point
  9. Tetrahedral subgroups
  10. Octahedral subgroups
  11. Icosahedral subgroups
  12. The -regular case
  13. Subgroups with elements of order
  14. The case
  15. The Case $q = 2^m$ with $m > 1$.
  16. ...and 10 more sections

Key Result

Proposition 3.1

Let $s \in \operatorname{PGL}_2(k)$. If $s$ fixes three distinct points of $\mathbb{P}^1(k)$, then $s$ is the identity.

Theorems & Definitions (80)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 70 more