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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.

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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