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On groups with two isomorphism classes of derived subgroups. (English) Zbl 1287.20046
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 F. de Giovanni and D. J. S. Robinson, [J. Lond. Math. Soc., II. Ser. 71, No. 3, 658-668 (2005; Zbl 1084.20026)], and M. Herzog, P. Longobardi and 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.
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.

##### MSC:
 20F14 Derived series, central series, and generalizations for groups 20E34 General structure theorems for groups 20F16 Solvable groups, supersolvable groups 20F18 Nilpotent groups
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