Let K be a compact metrizable group and Gamma be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Gamma implies Gamma contains ergodic automorphisms if center of the action, Z(Gamma) = {alpha is an element of Aut(K)vertical bar alpha commutes with elements of Gamma} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc. 289(1) (1985), 393-407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480-496].
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