On groups with two isomorphism classes of derived subgroups. (Q2845580)

If \(H\) is a subgroup of a group \(G\) then \(H\) is a derived subgroup of \(G\) if \(H=K'\) for some subgroup \(K\) of \(G\). Let \(\mathfrak C_n\) denote the class of groups in which there are at most \(n\) derived subgroups and let \(\mathfrak C\) denote the union of the classes \(\mathfrak C_n\). These classes have recently been investigated by \textit{F. de Giovanni} and \textit{D. J. S. Robinson}, [J. Lond. Math. Soc., II. Ser. 71, No. 3, 658-668 (2005; Zbl 1084.20026)], and \textit{M. Herzog, P. Longobardi} and \textit{M. Maj} [Contemp. Math. 402, Isr. Math. Conf. Proc. 181-192 (2006; Zbl 1122.20017)]. By contrast this paper is concerned with the groups in which the set of isomorphism types of derived subgroups is small.NEWLINENEWLINE If \(n\) is a positive integer, let \(\mathfrak D_n\) denote the class of groups whose derived subgroups fall into at most \(n\) isomorphism classes. Obviously, \(\mathfrak D_1=\mathfrak C_1\) is the class of Abelian groups and in this paper the class \(\mathfrak D_2\) is discussed. Thus a non-Abelian group \(G\in\mathfrak D_2\) if and only if \(H'\cong G'\) whenever \(H\) is a non-Abelian subgroup of \(G\). There are many diverse examples of \(\mathfrak D_2\)-groups. Among the many interesting theorems we give the flavor with two of these: Theorem 1: A non-Abelian group \(G\) is nilpotent and belongs to \(\mathfrak D_2\) if and only if \(G'\) is cyclic of prime or infinite order and \(G'\leq Z(G)\); Theorem 3 (abridged): Let \(G\) be a non-nilpotent soluble \(\mathfrak D_2\)-group. Then \(G\) is metabelian; \(G'\) is an elementary Abelian \(p\)-group for some prime \(p\), or a free Abelian group, or a torsion-free minimax group; and the nilpotent subgroups of \(G\) are Abelian.