An Example of a Nonlinearizable Quasicyclic Subgroup in the Automorphism Group of the Polynomial Algebra

As is well known, every finite subgroup of the automorphism group of the polynomial algebra of rank two over a field of characteristic zero is conjugate to the subgroup of linear automorphisms. We show that this can fail for an arbitrary periodic subgroup. We construct an example of an Abelian $p$-subgroup of the automorphism group of the polynomial algebra of rank two over the field of complex numbers which is not conjugate to any subgroup of linear automorphisms.