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In the paper, we consider the full group $[\phi]$ and topological full group$[[\phi]]$ of a Cantor minimal system $(X,\f)$. We prove that the commutatorsubgroups $D([\f])$ and $D([[\f]])$ are simple and show that the groups$D([\f])$ and $D([[\f]])$ completely determine the class of orbit equivalenceand flip conjugacy of $\f$, respectively. These results improve theclassification found in \cite{gps:1999}. As a corollary of the technique used,we establish the fact that $\f$ can be written as a product of threeinvolutions from $[\f]$.

Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems