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

TL;DR

This paper presents a new, explicit, and CFSG-free version of Jordan's theorem for finite subgroups of general linear groups over any field, extending previous results with quantitative bounds.

Contribution

It provides the first proof of Jordan's theorem that is explicit, CFSG-free, and applicable to arbitrary fields, using new computational techniques and dimensional estimates.

Findings

First CFSG-free, explicit Jordan's theorem over any field

Quantitative bounds on finite subgroups of GL(n,K)

Extension of previous methods with new computational techniques

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.