On Groups with Two Isomorphism Classes of Derived Subgroups
The structure of groups which have at most two isomorphism classes of derived subgroups (D-2-groups) is investigated. A complete description of D-2-groups is obtained in the case where the derived subgroup is finite: the solution leads an interesting number theoretic problem. In addition, detailed information is obtained about soluble D-2-groups, especially those with finite rank, where algebraic number fields play an important role. Also, detailed structural information about insoluble D-2-groups is found, and the locally free D-2-groups are characterized.
Glasgow Mathematical Journal
Longobardi, Patrizia; Maj, Mercede; Robinson, Derek J.S.; and Smith, Howard. "On Groups with Two Isomorphism Classes of Derived Subgroups." Glasgow Mathematical Journal (2013) : 655-668.