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

On the centralizers of minimal aperiodic actions on the Cantor set