Translations in certain groups of affine motions

The purpose of this article is to prove the conjecture of L. Auslander that every nilpotent group of affine motions of ${{\mathbf{R}}^n}$ that is simply transitive on ${{\mathbf{R}}^n}$ has a nontrivial translation in its center. The preliminary result that every such group is unipotent is of independent interest.