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 <!-- MATH ${{\mathbf{R}}^n}$ --> that is simply transitive on <!-- MATH ${{\mathbf{R}}^n}$ --> has a nontrivial translation in its center. The preliminary result that every such group is unipotent is of independent interest.