Groups in which the Derived Groups of all 2-generator Subgroups are Cyclic

Let P be a group theoretical property. There are many results in literature concerning groups in which every 2-generator subgroup has P, the main question being when in this condition the whole group G is in P. For example a finite group is soluble if every 2-generator subgroup is soluble, a finitely generated soluble group is nilpotent if every 2-generator subgroup is nilpotent. In this paper we consider the class C of groups for which the commutator ’ is cyclic, for all x,y in G. Groups with this property appear to have been first studied by J.L. Alperin. He proved that a finite nilpotent group in the class C is metabelian, and asked whether the restriction on the order is necessary. In this paper we first study finite groups in the class C, and, aswering to the Alperin’s question, we exhibit a finite 2-group in C which is not metabelian. We also generalise the above result by Alperin by proving that any finite group of odd order in the class C is metabelian. Then we study infinite groups in C. We prove that a torsion-free nilpotent group in the class C is metabelian and we obtain a complete description of torsion-free groups in C. We prove that there exists a torsion-free group G that lies in the class C but is not metabelian; furthermore G has a homomorphic image of order 2^{10} that is not metabelian (but obviously is in C). In the periodic case we show that any p-group with p odd in C is nilpotent, while if G is a 2-group in C, then G^2 is hypercentral. Finally we present a description of locally graded groups in C