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In commutative invariant theory, a classical result due to Auslander saysthat if $R = \Bbbk[x_1, \dots, x_n]$ and $G$ is a finite subgroup of$\text{Aut}_{\text{gr}}(R) \cong \text{GL}(n,\Bbbk)$ which contains noreflections, then there is a natural graded isomorphism $R \hspace{1pt} \#\hspace{1pt} G \cong \text{End}_{R^G}(R)$. In this paper, we show that aversion of Auslander's Theorem holds if we replace $R$ by an Artin-Schelterregular algebra $A$ of global dimension 2, and $G$ by a finite subgroup of$\text{Aut}_{\text{gr}}(A)$ which contains no quasi-reflections. This extendswork of Chan-Kirkman-Walton-Zhang. As part of the proof, we classify all suchpairs $(A,G)$, up to conjugation of $G$ by an element of$\text{Aut}_{\text{gr}}(A)$. In all but one case, we also write down explicitpresentations for the invariant rings $A^G$, and show that they are isomorphicto factors of AS regular algebras.

Actions of Small Groups on Two-Dimensional Artin-Schelter Regular Algebras