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On groups with finitely many derived subgroups. (English) Zbl 1303.20044
If $$G$$ is a locally graded group for which the set of derived subgroups is finite, it was proved by D. J. S. Robinson and the reviewer [J. Lond. Math. Soc., II. Ser. 71, No. 3, 658-668 (2005; Zbl 1084.20026)] that the derived subgroup $$G'$$ of $$G$$ is finite. Recall here that a group $$G$$ is said to be locally graded if every finitely generated non-trivial subgroup of $$G$$ contains a proper subgroup of finite index.
In the paper under review, the author obtains some new informations on the structure of such groups. In fact, he proves that if $$G$$ is a locally graded group with only $$k$$ derived subgroups (where $$k$$ is a positive integer), then the derived subgroup $$G'$$ of $$G$$ contains a nilpotent-by-Abelian subgroup with index at most $$k!$$. Moreover, he shows that any locally graded group with at most $$22$$ derived subgroups must be soluble, but the alternating group $$A_5$$ has precisely $$23$$ derived subgroups.

##### MSC:
 20F14 Derived series, central series, and generalizations for groups 20E34 General structure theorems for groups 20E25 Local properties of groups 20E07 Subgroup theorems; subgroup growth
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##### References:
 [1] DOI: 10.1112/S0024610705006484 · Zbl 1084.20026 [2] Dickson L. E., Linear Groups: With an Exposition of the Galois Field Theory (1958) · Zbl 0082.24901 [3] DOI: 10.1090/S0002-9939-1958-0093542-8 [4] DOI: 10.1090/conm/402/07578 [5] DOI: 10.1017/S0017089512000821 · Zbl 1287.20046 [6] DOI: 10.1090/S0002-9904-1968-11953-6 · Zbl 0159.30804
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