Almost-simple affine difference algebraic groups

TL;DR

This paper extends classical group theory results to affine difference algebraic groups, establishing analogs of isomorphism theorems and a Jordan-Hölder type decomposition for these generalized groups.

Contribution

It introduces a Jordan-Hölder type theorem for affine difference algebraic groups and characterizes almost-simple groups within this framework.

Findings

Established isomorphism theorems for affine difference algebraic groups.

Proved a Jordan-Hölder type decomposition theorem.

Characterized almost-simple affine difference algebraic groups via algebraic groups.

Abstract

Affine difference algebraic groups are a generalization of affine algebraic groups obtained by replacing algebraic equations with algebraic difference equations. We show that the isomorphism theorems from abstract group theory have meaningful analogs for these groups and we establish a Jordan-H\"{o}lder type theorem that allows us to decompose any affine difference algebraic group into almost-simple affine difference algebraic groups. We also characterize almost-simple affine difference algebraic groups via almost-simple affine algebraic groups.