In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal $ \mathbb Z $ action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group $ G $, the centralizer of any minimal aperiodic $ G $-action on the Cantor set is a subgroup of the centralizer of a minimal $ \mathbb Z $-action.
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