  In the paper we discuss the algebraic structure of topological full group
$[[T]]$ of a Cantor minimal system $(X,T)$. We show that the topological full
group $[[T]]$ has the structure similar to a union of permutational wreath
products of group $\mathbb Z$. This allows us to prove that the topological
full groups are locally embeddable into finite groups; give an elmentary proof
of the fact that group $[[T]]'$ is infinitely presented; and provide explicit
examples of maximal locally finite subgroups of $[[T]]$. We also show that the
commutator subgroup $[[T]]'$, which is simple and finitely-generated for
minimal subshifts, is decomposable into a product of two locally finite groups
and that the groups $[[T]]$ and $[[T]]'$ possess continuous ergodic invariant
random subgroups.
