ABSTRACT
We study outer automorphisms of ordered nilpotent groups which preserve the order of G. We denote the group of all automorphisms of a group G by Aut G, the group of all automorphisms of a group G which preserve the order by Ord G and the group of all automorphisms of a group G which act identically on the abelianization G/G' by IA G. In general, Aut G is not orderable and Ord G ф Aut G. In order to construct an ordered group G with two properties: (1) Aut G is orderable and (2) Aut G = Ord G, we study the case of ordered nilpotent groups. According to the definition of ordered group (see [1]) every inner automorphism preserves the order of the group. It is well known (see [2], Lemma 4.7) that every IA-automorphism acts identically on each factor of the descending central series. It follows that if the subgroups of the lower series are convex then every IA-automorphism preserves the order. So we concentrate our attention on torsion-free nilpotent groups with the property Aut G = IA G = Ord G.
