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Groups with finitely many derived subgroups. (English) Zbl 1084.20026
In this very nice paper the authors consider the classes $$\mathcal C$$ (respectively $$\mathcal C_\infty$$) of groups $$G$$ for which the set $$\{H'\mid H\leq G\}$$ (respectively $$\{H'\mid H\leq G$$, $$H$$ is infinite}) is finite. It is clear that finite-by-Abelian groups satisfy $$\mathcal C$$ and that $$\mathcal C$$ contains all groups of Tarski type (i.e. infinite simple groups whose proper subgroups are Abelian).
The authors prove a number of interesting results. For example a locally graded group with the property $$\mathcal C$$ is finite-by-Abelian. More generally (Theorem B) if $$G$$ is a group with property $$\mathcal C$$ there is a characteristic series $$G=S_0\geq G_1\geq S_1\geq\cdots\geq S_{t-1}\geq G_t=1$$ such that $$S_i/G_{i+1}$$ is finite-by-Abelian, $$G_i/S_i$$ is a simple group of Tarski type and $$G_i$$ is a finitely generated perfect group for all $$i$$.
Analogous results hold for the more general class of $$\mathcal C_\infty$$-groups. This is a wider class, as the locally dihedral group shows. However the locally dihedral group is in a sense typical of the kind of Chernikov group that can arise.

##### MSC:
 20F24 FC-groups and their generalizations 20E34 General structure theorems for groups 20F14 Derived series, central series, and generalizations for groups 20E25 Local properties of groups 20E32 Simple groups 20E07 Subgroup theorems; subgroup growth
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