Groups in Which the Commutator Subgroup is Cyclic

Abstract

So far, our attention was focused on finite groups in which the commutator is in the center and it’s called by the class of CC-groups. Since the center of any group is an abelian group, and the fundamental theorem of finitely generated abelian groups asserts that every abelian group is isomorphic to the direct product of cyclic groups. Then it is reasonable to consider those groups for which the derived subgroup is cyclic. It should be remarked that several authors have investigated particular classes of groups with similar restrictions. For instance, in [1] a bound is obtained for the order of $$G/G{^{\prime}}$$ when $$G$$ is a $$p-group$$ , $$p\ne 2$$ , with a cyclic commutator subgroup. In this paper, we prove that every finite group which has a cyclic commutator must be supersolvable. Our result makes it possible to apply all properties of supersolvable groups to the so-called $$Dc$$ -groups.