On Groups which contain no HNN-Extensions

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. 23 We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 25 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine the structure of locally graded groups in which 27 every subgroup is pronormal, thus generalizing a theorem of Kuzennyi and Subbotin.