# On extensions of algebraic groups

## TL;DR

This paper generalizes a classic group extension result to algebraic groups, relating extension classes to the center of the group, and applies it to finiteness over finite fields.

## Contribution

It extends known extension classification results from abstract groups to algebraic groups and analyzes split extensions with a finiteness application.

## Key findings

- Extension classes relate to the center of algebraic groups.
- Finiteness of split extension classes over finite fields.
- Generalizable proof approach.

## Abstract

We extend to the context of algebraic groups a classic result on extensions of abstract groups relating the set of isomorphism classes of extensions of $G$ by $H$ with that of extensions of $G$ by the center $Z$ of $H$. The proof should be easily generalizable to other contexts. We also study the subset of classes of split extensions and give a quick application by proving a finiteness result on these sets over a finite field.