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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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