# Definable Groups in DCFA

## TL;DR

This paper investigates the structure of definable groups within models of DCFA, showing they embed into algebraic groups and exploring properties like 1-basedness and stability.

## Contribution

It establishes that definable groups in DCFA embed into algebraic groups and analyzes their model-theoretic properties, advancing understanding of their structure.

## Key findings

- Definable groups embed into algebraic groups.
- Analysis of 1-basedness and stability of abelian definable groups.
- Insights into stable embeddability in DCFA.

## Abstract

E. Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion. We denote it DCFA. In this paper we study definable groups in a model of DCFA. First we prove that such a group is embeds on an algebraic group. Then we study 1-basedeness, stability and stable embeddability of abelian definable groups.