ABSTRACT
Thus we may strengthening the action of Aut G on pG and say that G is a U-group if any two automorphisms <p,tp' € AutG with gtp — gtp' for some g e pG must be the same <p = <p'. Hence G is a UT-group if and only if G is both a T-group as well as a U-group. Thus G is a UT-group if and only if Aut G is transitive and (every non-trivial automorphism acts) fix point-free on pG. In connection with permutation groups such action is also called 'sharply transitive'. Note that pG may be empty, if G is divisible for instance. In order to avoid trivial cases we also require that 0 ^ G ^ Z and G is of type 0, hence G is torsion-free and every element of G is a multiple of an element in pG. If G is of type 0 and not finitely generated then |pG| = |G| is large and the problem about the existence of UT-groups becomes really interesting. This question is related to problems in homotopy theory and was raised by Dror Farjoun. In response we want to show the following Theorem 1.1. For any successor cardinal X — p,+ with p, — p?° there is an Hi-/ree abelian UT-group of cardinality A.
