  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 commutator
subgroups $D([\f])$ and $D([[\f]])$ are simple and show that the groups
$D([\f])$ and $D([[\f]])$ completely determine the class of orbit equivalence
and flip conjugacy of $\f$, respectively. These results improve the
classification 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 three
involutions from $[\f]$.
