Bounds for the orders of the finite subgroups of G(k)

TL;DR

This paper establishes bounds on the sizes of finite subgroups within the rational points of reductive algebraic groups over fields, depending on the group's type and cyclotomic Galois groups.

Contribution

It provides new bounds for finite subgroup orders in G(k), linking algebraic group structure with Galois theory of cyclotomic extensions.

Findings

Bounds depend on the type of G and Galois groups of cyclotomic extensions

Results apply to reductive algebraic groups over fields

Provides a framework for understanding finite subgroup structures

Abstract

If k is a commutative field and G a reductive (connected) algebraic group over k, we give bounds for the orders of the finite subgroups of G(k); these bounds depends on the type of G and on the Galois groups of the cyclotomic extensions of k.