Ordered Groups in which Every Automorphism Preserves the Order
by
Vasiliy Bludov
Irkutsk State University

In the report we represent an example of orderable nilpotent group G with the property: every automorphism of G preserve every full order of G.

We use the following notations: Aut G - group of all automorphisms of G, IA G - subgroup of IA-automorphisms (i.e. subgroup of automorphisms induce identical automorphism on G/G').

Example. Let k, q - nonzero integers and G_{k, q} - nilpotent group of class 5 with generators a, b and defining relations:

[a, [a, [a, [a, b]]]]=[a, [a, b, b, b]]=[a, b, b, b, b]=1, [a, [a, [a, b, b]]]=[a, [a, b], [a, b]], [a, [a, [a, b]]]=[a, [a, b], [a, b]]k=1, [a, b, b, b]=[a, b, [a, b, b]]q.

Direct calculations show that G_{k, q} is orderable, Aut G_{k, q}=IA G_{k, q} and every automorphism of G_{k, q} preserve every full order of G_{k, q}. Moreover, IA G_{k, q} is orderable as torsion-free nilpotent group.

Remark. If G is a two-generated torsion-free group of nilpotency class less than 5 then Aut G has nontrivial elements of finite orders.

Date received: April 30, 1998