# On the characterization of Danielewski surfaces by their automorphism group

## TL;DR

This paper proves that a normal affine surface with an automorphism group isomorphic to that of a Danielewski surface must itself be a Danielewski surface, providing a characterization based on automorphism groups.

## Contribution

It establishes a group-theoretic characterization of Danielewski surfaces, showing that their automorphism groups uniquely determine their structure.

## Key findings

- Automorphism group isomorphism implies surface is Danielewski surface
- Provides a new criterion for identifying Danielewski surfaces
- Enhances understanding of automorphism groups of affine surfaces

## Abstract

In this note we show that if the automorphism group of a normal affine surface $S$ is isomorphic to the automorphism group of a Danielewski surface, then $S$ is isomorphic to a Danielewski surface.