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A CFSG-free explicit Jordan's theorem over arbitrary fields

Jitendra Bajpai, Daniele Dona

TL;DR

This work delivers a fully explicit, CFSG-free version of Jordan's theorem for finite subgroups of GL_n(K) over arbitrary fields. Building on Larsen–Pink's framework, it develops explicit dimensional and degree estimates, identifies an underlying finite field via unipotent elements, and extracts a simple finite group of Lie type as G^F through a Steinberg endomorphism. The authors then descend through quotients by the unipotent radical and the center to assemble a global description of Γ as a near-product of almost simple factors, yielding a normal series with a small index and well-understood factor structure. The result advances a long-standing problem by providing computable bounds and a strategy applicable to diameter estimates and subgroup analysis in GL_n without invoking CFSG.

Abstract

We prove a version of Jordan's classification theorem for finite subgroups of $\mathrm{GL}_{n}(K)$ that is at the same time quantitatively explicit, CFSG-free, and valid for arbitrary $K$. This is the first proof to satisfy all three properties at once. Our overall strategy follows Larsen and Pink [24], with explicit computations based on techniques developed by the authors and Helfgott [2, 3], particularly in relation to dimensional estimates.

A CFSG-free explicit Jordan's theorem over arbitrary fields

TL;DR

This work delivers a fully explicit, CFSG-free version of Jordan's theorem for finite subgroups of GL_n(K) over arbitrary fields. Building on Larsen–Pink's framework, it develops explicit dimensional and degree estimates, identifies an underlying finite field via unipotent elements, and extracts a simple finite group of Lie type as G^F through a Steinberg endomorphism. The authors then descend through quotients by the unipotent radical and the center to assemble a global description of Γ as a near-product of almost simple factors, yielding a normal series with a small index and well-understood factor structure. The result advances a long-standing problem by providing computable bounds and a strategy applicable to diameter estimates and subgroup analysis in GL_n without invoking CFSG.

Abstract

We prove a version of Jordan's classification theorem for finite subgroups of that is at the same time quantitatively explicit, CFSG-free, and valid for arbitrary . This is the first proof to satisfy all three properties at once. Our overall strategy follows Larsen and Pink [24], with explicit computations based on techniques developed by the authors and Helfgott [2, 3], particularly in relation to dimensional estimates.

Paper Structure

This paper contains 21 sections, 28 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Outline of the strategy
  3. Varieties
  4. Basic nomenclature
  5. Degrees
  6. Degree of intersections
  7. Linear algebraic groups
  8. Definition and basic properties
  9. Escaping from a subgroup
  10. Quotients
  11. Almost simple groups
  12. Almost simple and adjoint groups
  13. Classification of almost simple adjoint groups
  14. Weyl groups
  15. Steinberg endomorphisms
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a finite subgroup of $\mathrm{GL}_{n}(\mathbb{C})$. Then there is a normal abelian subgroup $A\unlhd\Gamma$ of index bounded by a constant $J(n)$ depending only on $n$.

Theorems & Definitions (55)

  • Theorem 1.1: Jordan's theorem
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 45 more