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Let $K$ be a field, and let $\Aut \,K^2$ be the group of polynomialautomorphisms of $K^2$. We investigate which subgroups are linear or not. Incharacteristic zero, there are small nonlinear subgroups and some big linearsubgroups. When $K$ has finite characteristic, the whole group $\Aut\,K^2$ islinear whenever $K$ is finite, and nonlinear otherwise.

Linearity and Non-linearity of Groups of Polynomial Automorphisms of $K^2$