On Semiabelian Groups

Recall that a group is called semiabelian if it is generated by its normal cyclic subgroups [6]. The class of semiabelian groups is a very natural generalization of the wellknown class of Dedekind groups (the groups in which all cyclic subgroups are normal). In the paper [6] Venzke showed that these groups could play a major role in the theory of supersoluble finite groups. Based on the notion of semiabelian group, he developed a theory of finite supersoluble group similar to the theory of finite nilpotent group. He also proved that finite semiabelian groups are nilpotent of class 2, closed under homomorphic images and direct products, and are not closed under subgroups or normal subgroups. In this paper we study semiabelian (finite and infinite) groups under some restrictions. In particular, the torsion-free semiablian groups and two-generated semiabelian groups are described.