  We find new presentations for the Thompson's groups $F$, the derived group
$F^{'}$ and the intermediate group $D$. These presentations have a common
ground in that their relators are the same and only the generating sets differ.
As an application of these presentations we extract the following consequences:
the cost of the group $F^{'}$ is 1 hence the cost cannot decide the
(non)amenability question of $F$; the $II_1$ factor $L(F^{'})$ is inner
asymptotically abelian and the reduced $C^*$-algebra of $F$ is not residually
finite dimensional.
